Q.

**Statement – 1**: For each natural number $$ n,(n+1)^{7}-n^{7}-1 \text { is divisible by } 7 \text { . } $$**Statement – 2**: For each natural number $n, n^{7}-n$ is divisible by 7 (1) Statement-1 is false, statement-2 is true. (2) Statement-1 is true, statement-2 is true; Statement-2 is correct explanation for statement-1. (3) Statement-1 is true, statement-2 is true; Statement-2 is not a correct explanation for statement-1. (4) Statement-1 is true, statement-2 is false.**[AIEEE-2011]****Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...**

**Sol.**(2)

**Statement-2:**$P(n)=n^{7}-n$ Put $n=1, \quad 1-1=0$ is divisible by 7 Let $n=k, \quad P(k)=k^{7}-k$ is divisible by 7 Put $n=k+1$ $\mathrm{P}(\mathrm{k}+1)=(\mathrm{k}+1)^{7}-(\mathrm{k}+1)$ $=\mathrm{k}^{7}+^{7} \mathrm{C}_{1} \mathrm{k}^{6}+\ldots \ldots+^{7} \mathrm{C}_{6} \mathrm{k}+1-\mathrm{k}-1$ $=\left(\mathrm{k}^{7}-\mathrm{k}\right)+$ multiple of 7 $\Rightarrow \mathrm{P}(\mathrm{k}+1)$ is divisible by 7 Hence $\mathrm{P}(\mathrm{n})=\mathrm{n}^{7}-\mathrm{n}$ is divisible by 7 Now statement- 1 $\mathrm{P}(\mathrm{n})=(\mathrm{n}+1)^{7}-\mathrm{n}^{7}-1$ $(\mathrm{n}+1)^{7}=(\mathrm{R}+1)^{7}-\left(\mathrm{n}^{7}-1\right.$ $\quad$ divisible by $7 \quad$ is divisible by 7 $\Rightarrow \mathrm{P}(\mathrm{n})=(\mathrm{n}+1)^{7}-\mathrm{n}^{7}-1$ is divisible by 7

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