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Q.
The sum to $n$ terms of the series $1+2\left(1+\frac{1}{n}\right)+3\left(1+\frac{1}{n}\right)^{2}+\ldots$ is given by
Sequences and Series
Solution:
Let $S$ be the sum of $n$ terms of the given series and $x=1+1 / n$.
Then,
$S=1+2 x+3 x^{2}+4 x^{3}+\ldots+ n x^{n-1}$
$\Rightarrow x S=x+2 x^{2}+3 x^{3}+\ldots+(n-1) x^{n-1}+n x^{n}$
$\therefore S-x S=1+\left[x+x^{2}+\ldots+x^{n-1}\right]-n x^{n}$
$\Rightarrow S(1-x)=\frac{1-x^{n}}{1-x}-n x^{n}$
$\Rightarrow S(-1 / n)=-n\left(1-(1+1 / n)^{n}\right)-n(1+1 / n)^{n}$
$\Rightarrow \frac{1}{n} S=n\left[1-(1+1 / n)^{n}+(1+1 / n)^{n}\right]=n$
$\Rightarrow S=n^{2}$