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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]
Ans. (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