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UPSC NDA Math Model Paper 5

Question : 58 of 120

Marks: +1, -0

\lim\limits_{x \to 1} \frac{x+x^{2}+x^{3}+\cdots+x^{n}-n}{x-1}

=5050

n

equals

Solution:

Let L =

\lim\limits_{x \to 1} \frac{x+x^{2}+x^{3}+\cdots+x^{n}-n}{x-1}

[Form of limit (0/0)]

Apply L-Hospital rule, we get

=\lim\limits_{x \to 1} \frac{1+2x+3x^{2}+4x^{3}+\cdots+n x^{n-1}-0}{1-0}

=1+2+3+\cdots+n=\frac{n(n+1)}{2}

Given:

L=5050

\Rightarrow \frac{n(n+1)}{2}=5050 \Rightarrow n^{2}+n=10100

\Rightarrow n^{2}+n-10100=0

\Rightarrow n^{2}+101n-100n-10100=0

\Rightarrow n(n+101)-100(n+101)=0

\Rightarrow (n-100)(n+101)=0 \therefore n=100

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