Question

For each natural number $n, (n + 1)^7 - n^7 -1$ is divisible by 7. For each natural number $n, n^7 - n$ is divisible by 7.

• A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement-1 is true, Statement-2 is false
• D. Statement-1 is false, Statement-2 is true

Answer 1

Answer: (A)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

Answer 2

The correct answer is A:Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
Given that;
Statement(S1): For each natural number ‘n’ $(n+1)^7-n^7-1$ is divisible by 7
Statement(S2): For each natural number n, $n^7-n$ is divisible by 7.
Let us use mathematical induction that can check statement 2 is true for $\forall n\in N$
$\therefore (n+1)^7-n^7-1=[(n+1)^7-(n+1)]-[n^7-n]$
Here both the terms are divisible by 7
$\therefore(n+1)^7-n^7-1$ is also divisible by 7
So, both these statements are true.