The sum of the first 10 terms of the series 5 1 ⋅ 2 ⋅ 3 + 7 2 ⋅ 3 ⋅ 9 + 9 3 ⋅ 4 ⋅ 27 + . . . . is

Question

The sum of the first 10 terms of the series 5 1 ⋅ 2 ⋅ 3 + 7 2 ⋅ 3 ⋅ 9 + 9 3 ⋅ 4 ⋅ 27 + . . . . is, 2 + 1 3 10, 1 + 1 11 3 10, 2 - 1 11 3 10, 1 - 1 11 3 10

MARKS App · Accepted Answer

T n = 2 n + 3 n n + 1 3 n 2 n + 3 n n + 1 = A n + B n + 1 ⇒ A = 3 , B = - 1 T n = 3 n - 1 n + 1 1 3 n = 1 n ⋅ 3 n - 1 - 1 n + 1 3 n = f n - f n + 1 where, f n = 1 n ⋅ 3 n - 1 ⇒ S n = T 1 + T 2 + . . . . . . + T n S 10 = T 1 + T 2 + . . . . . . + T 10 = f 1 - f 2 + f 2 - f 3 + . . . . . . + f 10 - f 11 S 10 = f 1 - f 11 = 1 1 ⋅ 3 0 - 1 11 ⋅ 3 10 = 1 - 1 11 ⋅ 3 10

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