Question

Statement 1 : For each natural number $n,(n+1{)}^7-n^7-1$ is divisible by 7.
Statement 2 : For each natural $n$, $n^7-n$ is divisible by 7.

• A. A) Statement - 1 is True, Statement - 2 is True; Statement -2 is a correct explanation for Statement - 1
• B. B) Statement - 1 is True, Statement - 2 is True; Statement -2 is NOT a correct explanation for Statement - 1
• C. C) Statement - 1 is True, Statement-2 is False
• D. D) Statement - 1 is False, Statement-2 is True

Answer 1

The correct option is A A) Statement - 1 is True, Statement - 2 is True; Statement -2 is a correct explanation for Statement - 1

It can be checked by using the Principle of mathematical induction that statement - 2 is true for all $n\in N.$

So, statement - 2 is correct.

Now, $(n+1{)}^7-n^7-1=\left\{\right(n+1{)}^7-(n+1)\}-\{n^7-n\}$

Here both the terms are divisible by $7.$

Therefore, $(n+1{)}^7-n^7-1$ is also divisible by $7$.

So, statement - 1 is correct and statement-2 is a correct explanation for statement-1.