First slide

Arithmetico-Geometric Progression

Question

Sum to $n$ terms of the series $S = 1 + 2 (1 + \frac{1}{n}) + 3 {(1 + \frac{1}{n})}^{2} + \dots$ is given by

Moderate

A

$n^{2}$
B

${(n + 1)}^{2}$
C

$n (n + 1)$
D

none of these

Solution

Let $x = 1 + 1 / n .$ Then
$\begin{array}{l} S = 1 + 2 x + 3 x^{2} \dots + n x^{n - 1} \\ \Rightarrow x S = x + 2 x^{2} + \dots + (n - 1) x^{n - 1} + n x^{n} \end{array}$
Subtracting, we get
$\begin{array}{l} (1 - x) S = 1 + x + x^{2} + \dots + x^{n - 1} - n x^{n} \\ = \frac{1 - x^{n}}{1 - x} - n x^{n} \\ \Rightarrow (- \frac{1}{n}) S = (- n) [1 - {(1 + \frac{1}{n})}^{n}] - n {(1 + \frac{1}{n})}^{n} = - n \\ \Rightarrow S = n^{2} \end{array}$

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