Chapter 5: Arithmetic Progressions

Solution

a=2, d=5. a₁₀ = 2+9(5) = 47.

Solution

2,4,8,16 has ratios 2,2,2 (GP, not AP). Differences are 2,4,8 — not constant.

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{129 \times 129+1}{2} = 8385\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{146 \times 146+1}{2} = 10731\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{125 \times 125+1}{2} = 7875\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{19 \times 19+1}{2} = 190\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{50 \times 50+1}{2} = 1275\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{46 \times 46+1}{2} = 1081\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{73 \times 73+1}{2} = 2701\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{15 \times 15+1}{2} = 120\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{20 \times 20+1}{2} = 210\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{64 \times 64+1}{2} = 2080\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{198 \times 198+1}{2} = 19701\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{74 \times 74+1}{2} = 2775\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{16 \times 16+1}{2} = 136\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{13 \times 13+1}{2} = 91\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{11 \times 11+1}{2} = 66\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{89 \times 89+1}{2} = 4005\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{181 \times 181+1}{2} = 16471\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{34 \times 34+1}{2} = 595\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{57 \times 57+1}{2} = 1653\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{120 \times 120+1}{2} = 7260\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{143 \times 143+1}{2} = 10296\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{190 \times 190+1}{2} = 18145\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{111 \times 111+1}{2} = 6216\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{195 \times 195+1}{2} = 19110\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{71 \times 71+1}{2} = 2556\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{101 \times 101+1}{2} = 5151\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{147 \times 147+1}{2} = 10878\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{170 \times 170+1}{2} = 14535\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{48 \times 48+1}{2} = 1176\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{136 \times 136+1}{2} = 9316\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{180 \times 180+1}{2} = 16290\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{134 \times 134+1}{2} = 9045\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{197 \times 197+1}{2} = 19503\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{17 \times 17+1}{2} = 153\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{127 \times 127+1}{2} = 8128\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{175 \times 175+1}{2} = 15400\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{103 \times 103+1}{2} = 5356\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{33 \times 33+1}{2} = 561\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{88 \times 88+1}{2} = 3916\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{59 \times 59+1}{2} = 1770\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{169 \times 169+1}{2} = 14365\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{191 \times 191+1}{2} = 18336\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{95 \times 95+1}{2} = 4560\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{171 \times 171+1}{2} = 14706\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{39 \times 39+1}{2} = 780\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{117 \times 117+1}{2} = 6903\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{188 \times 188+1}{2} = 17766\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{157 \times 157+1}{2} = 12403\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{94 \times 94+1}{2} = 4465\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{35 \times 35+1}{2} = 630\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{140 \times 140+1}{2} = 9870\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{184 \times 184+1}{2} = 17020\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{187 \times 187+1}{2} = 17578\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{121 \times 121+1}{2} = 7381\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{45 \times 45+1}{2} = 1035\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{67 \times 67+1}{2} = 2278\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{43 \times 43+1}{2} = 946\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{166 \times 166+1}{2} = 13861\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{44 \times 44+1}{2} = 990\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{192 \times 192+1}{2} = 18528\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{91 \times 91+1}{2} = 4186\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{109 \times 109+1}{2} = 5995\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{14 \times 14+1}{2} = 105\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{87 \times 87+1}{2} = 3828\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{68 \times 68+1}{2} = 2346\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{154 \times 154+1}{2} = 11935\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{165 \times 165+1}{2} = 13695\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{112 \times 112+1}{2} = 6328\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{69 \times 69+1}{2} = 2415\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{163 \times 163+1}{2} = 13366\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{160 \times 160+1}{2} = 12880\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{61 \times 61+1}{2} = 1891\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{52 \times 52+1}{2} = 1378\)

Solution

Sum of first n natural numbers \(= \frac{n(n+1)}{2}\)

\(= \frac{41 \times 41+1}{2} = 861\)

Solution

d=12. a₅₄ = 3+53(12) = 639. a₅₄+132 = 771 = 3+(n−1)12 → n−1=64 → n=65.

Solution

a=8, d=−5. S₂₂ = 22/2[16+21(−5)] = 11[16−105] = 11(−89) = −979.

Solution

a+10d=38, a+15d=73 → 5d=35 → d=7. a=38−70=−32. a₃₁=−32+30(7)=178.

Solution

S = 100/2(1+100) = 50 × 101 = 5050.

Solution

Sₙ = n/2(a+l) → 400 = n/2(50) → n = 16.

Solution

x is the middle term: x = (2+26)/2 = 14.

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 8 + (12-1) \times 9\)

\(= 8 + 107 - 8\) ... simplifying gives \(107\)

Solution

\(a_n = a + (n-1)d\)

\(a_{10} = 6 + (10-1) \times 4\)

\(= 6 + 42 - 6\) ... simplifying gives \(42\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 15 + (27-1) \times 10\)

\(= 15 + 275 - 15\) ... simplifying gives \(275\)

Solution

\(a_n = a + (n-1)d\)

\(a_{17} = 8 + (17-1) \times 2\)

\(= 8 + 40 - 8\) ... simplifying gives \(40\)

Solution

\(a_n = a + (n-1)d\)

\(a_{8} = 4 + (8-1) \times 6\)

\(= 4 + 46 - 4\) ... simplifying gives \(46\)

Solution

\(a_n = a + (n-1)d\)

\(a_{16} = 11 + (16-1) \times 9\)

\(= 11 + 146 - 11\) ... simplifying gives \(146\)

Solution

\(a_n = a + (n-1)d\)

\(a_{18} = 10 + (18-1) \times 2\)

\(= 10 + 44 - 10\) ... simplifying gives \(44\)

Solution

\(a_n = a + (n-1)d\)

\(a_{29} = 18 + (29-1) \times 6\)

\(= 18 + 186 - 18\) ... simplifying gives \(186\)

Solution

\(a_n = a + (n-1)d\)

\(a_{7} = 10 + (7-1) \times 1\)

\(= 10 + 16 - 10\) ... simplifying gives \(16\)

Solution

\(a_n = a + (n-1)d\)

\(a_{5} = 1 + (5-1) \times 9\)

\(= 1 + 37 - 1\) ... simplifying gives \(37\)

Solution

\(a_n = a + (n-1)d\)

\(a_{26} = 14 + (26-1) \times 8\)

\(= 14 + 214 - 14\) ... simplifying gives \(214\)

Solution

\(a_n = a + (n-1)d\)

\(a_{17} = 20 + (17-1) \times 5\)

\(= 20 + 100 - 20\) ... simplifying gives \(100\)

Solution

\(a_n = a + (n-1)d\)

\(a_{17} = 19 + (17-1) \times -1\)

\(= 19 + 3 - 19\) ... simplifying gives \(3\)

Solution

\(a_n = a + (n-1)d\)

\(a_{16} = 14 + (16-1) \times 1\)

\(= 14 + 29 - 14\) ... simplifying gives \(29\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 16 + (25-1) \times -3\)

\(= 16 + -56 - 16\) ... simplifying gives \(-56\)

Solution

\(a_n = a + (n-1)d\)

\(a_{26} = 4 + (26-1) \times -5\)

\(= 4 + -121 - 4\) ... simplifying gives \(-121\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 8 + (9-1) \times 2\)

\(= 8 + 24 - 8\) ... simplifying gives \(24\)

Solution

\(a_n = a + (n-1)d\)

\(a_{23} = 19 + (23-1) \times 8\)

\(= 19 + 195 - 19\) ... simplifying gives \(195\)

Solution

\(a_n = a + (n-1)d\)

\(a_{30} = 10 + (30-1) \times -3\)

\(= 10 + -77 - 10\) ... simplifying gives \(-77\)

Solution

\(a_n = a + (n-1)d\)

\(a_{16} = 8 + (16-1) \times 10\)

\(= 8 + 158 - 8\) ... simplifying gives \(158\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 9 + (27-1) \times 3\)

\(= 9 + 87 - 9\) ... simplifying gives \(87\)

Solution

\(a_n = a + (n-1)d\)

\(a_{18} = 17 + (18-1) \times -1\)

\(= 17 + 0 - 17\) ... simplifying gives \(0\)

Solution

\(a_n = a + (n-1)d\)

\(a_{29} = 11 + (29-1) \times 6\)

\(= 11 + 179 - 11\) ... simplifying gives \(179\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 11 + (25-1) \times -5\)

\(= 11 + -109 - 11\) ... simplifying gives \(-109\)

Solution

\(a_n = a + (n-1)d\)

\(a_{18} = 15 + (18-1) \times -5\)

\(= 15 + -70 - 15\) ... simplifying gives \(-70\)

Solution

\(a_n = a + (n-1)d\)

\(a_{29} = 3 + (29-1) \times 8\)

\(= 3 + 227 - 3\) ... simplifying gives \(227\)

Solution

\(a_n = a + (n-1)d\)

\(a_{23} = 19 + (23-1) \times 0\)

\(= 19 + 19 - 19\) ... simplifying gives \(19\)

Solution

\(a_n = a + (n-1)d\)

\(a_{29} = 11 + (29-1) \times 7\)

\(= 11 + 207 - 11\) ... simplifying gives \(207\)

Solution

\(a_n = a + (n-1)d\)

\(a_{14} = 18 + (14-1) \times 7\)

\(= 18 + 109 - 18\) ... simplifying gives \(109\)

Solution

\(a_n = a + (n-1)d\)

\(a_{17} = 16 + (17-1) \times -5\)

\(= 16 + -64 - 16\) ... simplifying gives \(-64\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 2 + (25-1) \times 5\)

\(= 2 + 122 - 2\) ... simplifying gives \(122\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 4 + (25-1) \times -2\)

\(= 4 + -44 - 4\) ... simplifying gives \(-44\)

Solution

\(a_n = a + (n-1)d\)

\(a_{24} = 10 + (24-1) \times 9\)

\(= 10 + 217 - 10\) ... simplifying gives \(217\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 19 + (27-1) \times -4\)

\(= 19 + -85 - 19\) ... simplifying gives \(-85\)

Solution

\(a_n = a + (n-1)d\)

\(a_{28} = 1 + (28-1) \times -2\)

\(= 1 + -53 - 1\) ... simplifying gives \(-53\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 17 + (27-1) \times 0\)

\(= 17 + 17 - 17\) ... simplifying gives \(17\)

Solution

\(a_n = a + (n-1)d\)

\(a_{8} = 1 + (8-1) \times 3\)

\(= 1 + 22 - 1\) ... simplifying gives \(22\)

Solution

\(a_n = a + (n-1)d\)

\(a_{28} = 16 + (28-1) \times 5\)

\(= 16 + 151 - 16\) ... simplifying gives \(151\)

Solution

\(a_n = a + (n-1)d\)

\(a_{21} = 19 + (21-1) \times 9\)

\(= 19 + 199 - 19\) ... simplifying gives \(199\)

Solution

\(a_n = a + (n-1)d\)

\(a_{30} = 20 + (30-1) \times 7\)

\(= 20 + 223 - 20\) ... simplifying gives \(223\)

Solution

\(a_n = a + (n-1)d\)

\(a_{7} = 7 + (7-1) \times -4\)

\(= 7 + -17 - 7\) ... simplifying gives \(-17\)

Solution

\(a_n = a + (n-1)d\)

\(a_{5} = 5 + (5-1) \times 10\)

\(= 5 + 45 - 5\) ... simplifying gives \(45\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 11 + (25-1) \times 5\)

\(= 11 + 131 - 11\) ... simplifying gives \(131\)

Solution

\(a_n = a + (n-1)d\)

\(a_{30} = 17 + (30-1) \times 1\)

\(= 17 + 46 - 17\) ... simplifying gives \(46\)

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 3 + (12-1) \times 6\)

\(= 3 + 69 - 3\) ... simplifying gives \(69\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 12 + (9-1) \times -3\)

\(= 12 + -12 - 12\) ... simplifying gives \(-12\)

Solution

\(a_n = a + (n-1)d\)

\(a_{8} = 20 + (8-1) \times 9\)

\(= 20 + 83 - 20\) ... simplifying gives \(83\)

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 18 + (12-1) \times 4\)

\(= 18 + 62 - 18\) ... simplifying gives \(62\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 12 + (9-1) \times -2\)

\(= 12 + -4 - 12\) ... simplifying gives \(-4\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 14 + (25-1) \times 1\)

\(= 14 + 38 - 14\) ... simplifying gives \(38\)

Solution

\(a_n = a + (n-1)d\)

\(a_{11} = 4 + (11-1) \times 10\)

\(= 4 + 104 - 4\) ... simplifying gives \(104\)

Solution

\(a_n = a + (n-1)d\)

\(a_{26} = 7 + (26-1) \times 1\)

\(= 7 + 32 - 7\) ... simplifying gives \(32\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 8 + (9-1) \times 4\)

\(= 8 + 40 - 8\) ... simplifying gives \(40\)

Solution

\(a_n = a + (n-1)d\)

\(a_{5} = 12 + (5-1) \times -4\)

\(= 12 + -4 - 12\) ... simplifying gives \(-4\)

Solution

\(a_n = a + (n-1)d\)

\(a_{13} = 19 + (13-1) \times 6\)

\(= 19 + 91 - 19\) ... simplifying gives \(91\)

Solution

\(a_n = a + (n-1)d\)

\(a_{25} = 1 + (25-1) \times 9\)

\(= 1 + 217 - 1\) ... simplifying gives \(217\)

Solution

\(a_n = a + (n-1)d\)

\(a_{24} = 12 + (24-1) \times 9\)

\(= 12 + 219 - 12\) ... simplifying gives \(219\)

Solution

\(a_n = a + (n-1)d\)

\(a_{30} = 11 + (30-1) \times 0\)

\(= 11 + 11 - 11\) ... simplifying gives \(11\)

Solution

\(a_n = a + (n-1)d\)

\(a_{20} = 3 + (20-1) \times -4\)

\(= 3 + -73 - 3\) ... simplifying gives \(-73\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 17 + (27-1) \times -3\)

\(= 17 + -61 - 17\) ... simplifying gives \(-61\)

Solution

\(a_n = a + (n-1)d\)

\(a_{15} = 10 + (15-1) \times 2\)

\(= 10 + 38 - 10\) ... simplifying gives \(38\)

Solution

\(a_n = a + (n-1)d\)

\(a_{19} = 5 + (19-1) \times 2\)

\(= 5 + 41 - 5\) ... simplifying gives \(41\)

Solution

\(a_n = a + (n-1)d\)

\(a_{14} = 7 + (14-1) \times 0\)

\(= 7 + 7 - 7\) ... simplifying gives \(7\)

Solution

\(a_n = a + (n-1)d\)

\(a_{28} = 12 + (28-1) \times 4\)

\(= 12 + 120 - 12\) ... simplifying gives \(120\)

Solution

\(a_n = a + (n-1)d\)

\(a_{24} = 15 + (24-1) \times 2\)

\(= 15 + 61 - 15\) ... simplifying gives \(61\)

Solution

\(a_n = a + (n-1)d\)

\(a_{24} = 19 + (24-1) \times -5\)

\(= 19 + -96 - 19\) ... simplifying gives \(-96\)

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 13 + (12-1) \times 4\)

\(= 13 + 57 - 13\) ... simplifying gives \(57\)

Solution

\(a_n = a + (n-1)d\)

\(a_{27} = 19 + (27-1) \times 9\)

\(= 19 + 253 - 19\) ... simplifying gives \(253\)

Solution

\(a_n = a + (n-1)d\)

\(a_{28} = 3 + (28-1) \times 9\)

\(= 3 + 246 - 3\) ... simplifying gives \(246\)

Solution

\(a_n = a + (n-1)d\)

\(a_{22} = 2 + (22-1) \times 10\)

\(= 2 + 212 - 2\) ... simplifying gives \(212\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 1 + (9-1) \times 3\)

\(= 1 + 25 - 1\) ... simplifying gives \(25\)

Solution

\(a_n = a + (n-1)d\)

\(a_{21} = 9 + (21-1) \times 5\)

\(= 9 + 109 - 9\) ... simplifying gives \(109\)

Solution

\(a_n = a + (n-1)d\)

\(a_{22} = 14 + (22-1) \times -5\)

\(= 14 + -91 - 14\) ... simplifying gives \(-91\)

Solution

\(a_n = a + (n-1)d\)

\(a_{8} = 5 + (8-1) \times 4\)

\(= 5 + 33 - 5\) ... simplifying gives \(33\)

Solution

\(a_n = a + (n-1)d\)

\(a_{18} = 3 + (18-1) \times 9\)

\(= 3 + 156 - 3\) ... simplifying gives \(156\)

Solution

\(a_n = a + (n-1)d\)

\(a_{28} = 17 + (28-1) \times 0\)

\(= 17 + 17 - 17\) ... simplifying gives \(17\)

Solution

\(a_n = a + (n-1)d\)

\(a_{24} = 14 + (24-1) \times -3\)

\(= 14 + -55 - 14\) ... simplifying gives \(-55\)

Solution

\(a_n = a + (n-1)d\)

\(a_{9} = 2 + (9-1) \times 7\)

\(= 2 + 58 - 2\) ... simplifying gives \(58\)

Solution

\(a_n = a + (n-1)d\)

\(a_{15} = 5 + (15-1) \times -1\)

\(= 5 + -9 - 5\) ... simplifying gives \(-9\)

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 15 + (12-1) \times -3\)

\(= 15 + -18 - 15\) ... simplifying gives \(-18\)

Solution

\(a_n = a + (n-1)d\)

\(a_{15} = 3 + (15-1) \times -4\)

\(= 3 + -53 - 3\) ... simplifying gives \(-53\)

Solution

\(a_n = a + (n-1)d\)

\(a_{17} = 7 + (17-1) \times -5\)

\(= 7 + -73 - 7\) ... simplifying gives \(-73\)

Solution

\(a_n = a + (n-1)d\)

\(a_{30} = 20 + (30-1) \times 9\)

\(= 20 + 281 - 20\) ... simplifying gives \(281\)

Solution

\(a_n = a + (n-1)d\)

\(a_{14} = 8 + (14-1) \times -5\)

\(= 8 + -57 - 8\) ... simplifying gives \(-57\)

Solution

\(a_n = a + (n-1)d\)

\(a_{29} = 13 + (29-1) \times -2\)

\(= 13 + -43 - 13\) ... simplifying gives \(-43\)

Solution

\(a_n = a + (n-1)d\)

\(a_{12} = 15 + (12-1) \times 10\)

\(= 15 + 125 - 15\) ... simplifying gives \(125\)

Solution

\(a_n = a + (n-1)d\)

\(a_{15} = 1 + (15-1) \times 7\)

\(= 1 + 99 - 1\) ... simplifying gives \(99\)

Solution

\(a_n = a + (n-1)d\)

\(a_{6} = 1 + (6-1) \times 6\)

\(= 1 + 31 - 1\) ... simplifying gives \(31\)

Solution

\(a_n = a + (n-1)d\)

\(a_{7} = 13 + (7-1) \times 0\)

\(= 13 + 13 - 13\) ... simplifying gives \(13\)

Solution

a=9, d=8. Sₙ = n/2[18+(n−1)8] = n/2[8n+10] = n(4n+5).

4n²+5n=636 → 4n²+5n−636=0 → (4n+53)(n−12)=0. n=12.

Solution

a₄+a₈ = 2a+10d = 24 ... (i). a₆+a₁₀ = 2a+14d = 44 ... (ii).

Subtract: 4d=20 → d=5. 2a+50=24 → a=−13. AP: −13, −8, −3.

Solution

Sum div by 2: 2+4+...+100 = 50/2(2+100)=2550.

Sum div by 5: 5+10+...+100 = 20/2(5+100)=1050.

Sum div by 10: 10+20+...+100 = 10/2(10+100)=550.

By inclusion-exclusion: 2550+1050−550 = 3050.

Solution

a+2d=4, a+8d=−8 → 6d=−12 → d=−2. a=8. aₙ=0 → 8+(n−1)(−2)=0 → n=5.

Solution

a+11d=−13 ... (i). S₄ = 4/2(2a+3d) = 2(2a+3d) = 24 → 2a+3d=12 ... (ii).

From (i): a=−13−11d. Substitute: −26−22d+3d=12 → −19d=38 → d=−2. a=−13+22=9.

S₂₄ = 24/2[18+23(−2)] = 12[18−46] = 12(−28) = −336.

Wait, let me recheck: S₂₄ = 12[2(9)+23(−2)] = 12[18−46] = 12(−28) = −336.

Hmm, let me verify a₁₂: 9+11(−2) = 9−22 = −13 ✓. S₄ = 2[18+3(−2)] = 2(12) = 24 ✓.

S₂₄ = −336.

Solution

a₁₇−a₁₀ = 7d = 7 → d = 1.

Solution

AP: 11, 13, 15, ..., 99. a=11, d=2, l=99. n = (99−11)/2+1 = 45. S = 45/2(11+99) = 45×55 = 2475.

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{22} = \frac{22}{2}[2 \times 12 + (22-1) \times 4]\)

\(= 1188\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 2 + (11-1) \times 7]\)

\(= 407\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{23} = \frac{23}{2}[2 \times 13 + (23-1) \times 1]\)

\(= 552\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{6} = \frac{6}{2}[2 \times 9 + (6-1) \times 6]\)

\(= 144\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 8 + (7-1) \times 2]\)

\(= 98\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 9 + (25-1) \times 3]\)

\(= 1125\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 11 + (25-1) \times 3]\)

\(= 1175\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{6} = \frac{6}{2}[2 \times 14 + (6-1) \times 5]\)

\(= 159\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 6 + (25-1) \times 4]\)

\(= 1350\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 7 + (7-1) \times 1]\)

\(= 70\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{19} = \frac{19}{2}[2 \times 6 + (19-1) \times 8]\)

\(= 1482\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{6} = \frac{6}{2}[2 \times 1 + (6-1) \times 6]\)

\(= 96\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{17} = \frac{17}{2}[2 \times 12 + (17-1) \times 4]\)

\(= 748\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{5} = \frac{5}{2}[2 \times 13 + (5-1) \times 1]\)

\(= 75\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{17} = \frac{17}{2}[2 \times 7 + (17-1) \times 7]\)

\(= 1071\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{21} = \frac{21}{2}[2 \times 4 + (21-1) \times 1]\)

\(= 294\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{21} = \frac{21}{2}[2 \times 3 + (21-1) \times 7]\)

\(= 1533\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{12} = \frac{12}{2}[2 \times 15 + (12-1) \times 4]\)

\(= 444\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{18} = \frac{18}{2}[2 \times 14 + (18-1) \times 8]\)

\(= 1476\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 5 + (13-1) \times 2]\)

\(= 221\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{16} = \frac{16}{2}[2 \times 12 + (16-1) \times 5]\)

\(= 792\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 6 + (14-1) \times 8]\)

\(= 812\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 8 + (13-1) \times 6]\)

\(= 572\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{20} = \frac{20}{2}[2 \times 11 + (20-1) \times 3]\)

\(= 790\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 13 + (11-1) \times 3]\)

\(= 308\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 1 + (13-1) \times 8]\)

\(= 637\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{23} = \frac{23}{2}[2 \times 4 + (23-1) \times 6]\)

\(= 1610\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{20} = \frac{20}{2}[2 \times 5 + (20-1) \times 4]\)

\(= 860\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 10 + (7-1) \times 4]\)

\(= 154\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{10} = \frac{10}{2}[2 \times 14 + (10-1) \times 4]\)

\(= 320\)

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{10}{100}\right)^{1}\)

CI = A - P = 800.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{10}{100}\right)^{2}\)

CI = A - P = 3150.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{20}{100}\right)^{2}\)

CI = A - P = 2200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{5}{100}\right)^{1}\)

CI = A - P = 400.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{10}{100}\right)^{3}\)

CI = A - P = 331.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 300.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{12}{100}\right)^{2}\)

CI = A - P = 1272.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{5}{100}\right)^{3}\)

CI = A - P = 1261.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{5}{100}\right)^{3}\)

CI = A - P = 2364.38

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{20}{100}\right)^{3}\)

CI = A - P = 18200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 3895.68

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 2250.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{20}{100}\right)^{2}\)

CI = A - P = 440.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{5}{100}\right)^{3}\)

CI = A - P = 788.13

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{10}{100}\right)^{2}\)

CI = A - P = 4200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 2077.7

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{12}{100}\right)^{1}\)

CI = A - P = 120.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{20}{100}\right)^{2}\)

CI = A - P = 4400.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{5}{100}\right)^{3}\)

CI = A - P = 3152.5

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{20}{100}\right)^{3}\)

CI = A - P = 728.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{12}{100}\right)^{2}\)

CI = A - P = 508.8

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{15}{100}\right)^{3}\)

CI = A - P = 5208.75

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{20}{100}\right)^{3}\)

CI = A - P = 7280.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{5}{100}\right)^{2}\)

CI = A - P = 820.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{12}{100}\right)^{1}\)

CI = A - P = 960.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 3750.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{10}{100}\right)^{1}\)

CI = A - P = 500.0

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{22} = \frac{22}{2}[2 \times 5 + (22-1) \times 6]\)

\(= 1496\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{18} = \frac{18}{2}[2 \times 1 + (18-1) \times 1]\)

\(= 171\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 3 + (13-1) \times 7]\)

\(= 585\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 10 + (14-1) \times 5]\)

\(= 595\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{5} = \frac{5}{2}[2 \times 6 + (5-1) \times 1]\)

\(= 40\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{12} = \frac{12}{2}[2 \times 14 + (12-1) \times 6]\)

\(= 564\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{5} = \frac{5}{2}[2 \times 2 + (5-1) \times 7]\)

\(= 80\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{19} = \frac{19}{2}[2 \times 14 + (19-1) \times 6]\)

\(= 1292\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 7 + (11-1) \times 6]\)

\(= 407\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 5 + (14-1) \times 8]\)

\(= 798\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{18} = \frac{18}{2}[2 \times 4 + (18-1) \times 6]\)

\(= 990\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 6 + (14-1) \times 4]\)

\(= 448\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{22} = \frac{22}{2}[2 \times 10 + (22-1) \times 5]\)

\(= 1375\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{15} = \frac{15}{2}[2 \times 14 + (15-1) \times 8]\)

\(= 1050\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{23} = \frac{23}{2}[2 \times 12 + (23-1) \times 4]\)

\(= 1288\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{21} = \frac{21}{2}[2 \times 2 + (21-1) \times 2]\)

\(= 462\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{17} = \frac{17}{2}[2 \times 4 + (17-1) \times 6]\)

\(= 884\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{6} = \frac{6}{2}[2 \times 15 + (6-1) \times 1]\)

\(= 105\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{17} = \frac{17}{2}[2 \times 12 + (17-1) \times 3]\)

\(= 612\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 3 + (25-1) \times 5]\)

\(= 1575\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 14 + (7-1) \times 5]\)

\(= 203\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{20} = \frac{20}{2}[2 \times 5 + (20-1) \times 7]\)

\(= 1430\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 12 + (14-1) \times 2]\)

\(= 350\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{10} = \frac{10}{2}[2 \times 10 + (10-1) \times 7]\)

\(= 415\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{12} = \frac{12}{2}[2 \times 15 + (12-1) \times 6]\)

\(= 576\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{21} = \frac{21}{2}[2 \times 4 + (21-1) \times 4]\)

\(= 924\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{16} = \frac{16}{2}[2 \times 13 + (16-1) \times 5]\)

\(= 808\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{15} = \frac{15}{2}[2 \times 10 + (15-1) \times 5]\)

\(= 675\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 11 + (11-1) \times 3]\)

\(= 286\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{9} = \frac{9}{2}[2 \times 1 + (9-1) \times 6]\)

\(= 225\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{5} = \frac{5}{2}[2 \times 7 + (5-1) \times 7]\)

\(= 105\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{15} = \frac{15}{2}[2 \times 2 + (15-1) \times 3]\)

\(= 345\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{18} = \frac{18}{2}[2 \times 12 + (18-1) \times 7]\)

\(= 1287\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{16} = \frac{16}{2}[2 \times 4 + (16-1) \times 5]\)

\(= 664\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 3 + (13-1) \times 2]\)

\(= 195\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 13 + (11-1) \times 5]\)

\(= 418\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{20} = \frac{20}{2}[2 \times 15 + (20-1) \times 4]\)

\(= 1060\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{13} = \frac{13}{2}[2 \times 15 + (13-1) \times 4]\)

\(= 507\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 2 + (11-1) \times 1]\)

\(= 77\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{5} = \frac{5}{2}[2 \times 3 + (5-1) \times 8]\)

\(= 95\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{10} = \frac{10}{2}[2 \times 10 + (10-1) \times 4]\)

\(= 280\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{8} = \frac{8}{2}[2 \times 6 + (8-1) \times 6]\)

\(= 216\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{9} = \frac{9}{2}[2 \times 5 + (9-1) \times 6]\)

\(= 261\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{16} = \frac{16}{2}[2 \times 5 + (16-1) \times 8]\)

\(= 1040\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{11} = \frac{11}{2}[2 \times 1 + (11-1) \times 1]\)

\(= 66\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{19} = \frac{19}{2}[2 \times 15 + (19-1) \times 4]\)

\(= 969\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 14 + (14-1) \times 5]\)

\(= 651\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 8 + (7-1) \times 7]\)

\(= 203\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 9 + (14-1) \times 7]\)

\(= 763\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{20} = \frac{20}{2}[2 \times 9 + (20-1) \times 4]\)

\(= 940\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{8} = \frac{8}{2}[2 \times 3 + (8-1) \times 4]\)

\(= 136\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 14 + (25-1) \times 2]\)

\(= 950\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{14} = \frac{14}{2}[2 \times 8 + (14-1) \times 5]\)

\(= 567\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{23} = \frac{23}{2}[2 \times 7 + (23-1) \times 7]\)

\(= 1932\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{7} = \frac{7}{2}[2 \times 12 + (7-1) \times 1]\)

\(= 105\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 2 + (25-1) \times 1]\)

\(= 350\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{9} = \frac{9}{2}[2 \times 3 + (9-1) \times 1]\)

\(= 63\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{21} = \frac{21}{2}[2 \times 6 + (21-1) \times 7]\)

\(= 1596\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{25} = \frac{25}{2}[2 \times 4 + (25-1) \times 1]\)

\(= 400\)

Solution

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

\(S_{8} = \frac{8}{2}[2 \times 15 + (8-1) \times 6]\)

\(= 288\)

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{20}{100}\right)^{2}\)

CI = A - P = 3520.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{20}{100}\right)^{1}\)

CI = A - P = 200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{8}{100}\right)^{1}\)

CI = A - P = 2000.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{20}{100}\right)^{2}\)

CI = A - P = 880.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{12}{100}\right)^{1}\)

CI = A - P = 2400.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 519.42

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 5194.24

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 6492.8

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{12}{100}\right)^{1}\)

CI = A - P = 1200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{10}{100}\right)^{1}\)

CI = A - P = 2500.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{8}{100}\right)^{3}\)

CI = A - P = 259.71

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{8}{100}\right)^{2}\)

CI = A - P = 166.4

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 750.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{5}{100}\right)^{3}\)

CI = A - P = 1576.25

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{12}{100}\right)^{1}\)

CI = A - P = 600.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{20}{100}\right)^{1}\)

CI = A - P = 4000.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{12}{100}\right)^{2}\)

CI = A - P = 6360.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{12}{100}\right)^{2}\)

CI = A - P = 5088.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{15}{100}\right)^{2}\)

CI = A - P = 8062.5

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{5}{100}\right)^{1}\)

CI = A - P = 1000.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{8}{100}\right)^{1}\)

CI = A - P = 1600.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{15}{100}\right)^{3}\)

CI = A - P = 520.87

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{15}{100}\right)^{2}\)

CI = A - P = 4837.5

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{8}{100}\right)^{2}\)

CI = A - P = 3328.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{8}{100}\right)^{2}\)

CI = A - P = 832.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{5}{100}\right)^{2}\)

CI = A - P = 205.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 1500.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 8000\left(1 + \frac{15}{100}\right)^{1}\)

CI = A - P = 1200.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 2000\left(1 + \frac{15}{100}\right)^{2}\)

CI = A - P = 645.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 25000\left(1 + \frac{10}{100}\right)^{2}\)

CI = A - P = 5250.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 15000\left(1 + \frac{8}{100}\right)^{2}\)

CI = A - P = 2496.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 10000\left(1 + \frac{12}{100}\right)^{3}\)

CI = A - P = 4049.28

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 5000\left(1 + \frac{20}{100}\right)^{1}\)

CI = A - P = 1000.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 20000\left(1 + \frac{15}{100}\right)^{2}\)

CI = A - P = 6450.0

Solution

\(A = P\left(1 + \frac{r}{100}\right)^n\)

\(= 1000\left(1 + \frac{5}{100}\right)^{2}\)

CI = A - P = 102.5

Solution

a₁ = S₁ = 8. a₂ = S₂−S₁ = 22−8 = 14. d = 14−8 = 6. General: aₙ = Sₙ−Sₙ₋₁ = 3n²+5n−3(n−1)²−5(n−1) = 6n+2. Check: a₁=8, a₂=14 ✓.

Solution

Sₙ ratio: [n/2(2a₁+(n−1)d₁)] / [n/2(2a₂+(n−1)d₂)] = (7n+1)/(4n+27).

For 11th term ratio, put n=21 (since a₁₁ corresponds to the (n+1)/2 = 11th middle term when n=21):

(7×21+1)/(4×21+27) = 148/111 = 4/3.

Solution

S₆ = 3(2a+5d) = 42 → 2a+5d = 14 ... (i).

a₁₀/a₃₀ = (a+9d)/(a+29d) = 1/3 → 3a+27d = a+29d → 2a = 2d → a = d ... (ii).

From (i): 7d = 14 → d = 2, a = 2.

Solution

AP with 7 terms, d = −20, S₇ = 700.

700 = 7/2[2a+6(−20)] = 7/2(2a−120). 200 = 2a−120. a = 160.

Prizes: 160, 140, 120, 100, 80, 60, 40.

Solution

Since a,b,c are in AP: a+c = 2b.

a³+c³ = (a+c)(a²−ac+c²) = (a+c)[(a+c)²−3ac] = 2b[4b²−3ac].

LHS = 2b(4b²−3ac)+6abc = 8b³−6abc+6abc = 8b³ = RHS. ∎