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**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]**

**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

Excellent

Very bad