Statement -1 For each natural number `n,(n+1)^(7)-n^7-1` is divisible by 7.
Statement -2 For each natural number `n,n^7-n` is divisible by 7.

Options

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

Explanation

Let `P(n)=n^(7) -n`
By mathematical induction for `n=1, P(1)=0` , which is divisible by 7
for `n=k,P(k)=k^7-k`
Assume P(k) is divisible by 7
`therefore k^7-k=7lambda, lambda in I` .......(i)
For `n=k+1`.
`P(k+1)=(k+1)^7-(k+1)=(.^7C_0k^7+.^7C_1k^6+.^7C_2k^5+.^7C_3k^4+......+.^7C_6k+.^7C_7)-(k+1)`
`=(k^7-k)+7(k^6+3k^5+...+k)`
`=7lambda+7(k^6+3k^5+....+k)=` Divisible by 7
`therefore ` Statement -2 is true .
Also , let `F(n)=(n+1)^7-n^7-1`
`={(n+1)^7-(n+1)}-(n^7-n)`
=Divisible by 7 from Statement -2
Hence , both statements are true and Statement-2 is correct explanation of Statement-1.