First slide

Arithmetico-Geometric Progression

Question

The sum to n terms of the series $1 + 2 (1 + \frac{1}{n}) + 3 {(1 + \frac{1}{n})}^{2} + \dots . . .$ is given by

Moderate

A

n²
B

$n (n + 1)$
C

$n (1 + 1 / n)^{2}$
D

none of these

Solution

Let S be the sum of n terms of the given series and d x =1+1/n. Then,
$\begin{aligned} S = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots + {nx}^{n - 1} \\ \Rightarrow & xS = x + 2 x^{2} + 3 x^{3} + \dots + (n - 1) x^{n - 1} + {nx}^{n} \\ ∴ & S - xS = 1 + {x + x^{2} + \dots + x^{n - 1}} - {nx}^{n} \\ \Rightarrow & S (1 - x) = \frac{1 - x^{n}}{1 - x} - {nx}^{n} \\ \Rightarrow & S (- 1 / n) = - n [1 - (1 + 1 / n)^{n}] - n (1 + 1 / n)^{n} \\ \Rightarrow & \frac{1}{n} S = n {1 - (1 + 1 / n)^{n} + (1 + 1 / n)^{n}} = n \\ \Rightarrow & S = n^{2} \end{aligned}$

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