Statement -1 for all natural numbers n , `2.7^(n)+3.5^(n)-5` is divisible by 24.
Statement -2 if f(x) is divisible by x, then `f(x+1)-f(x)` is divisible by `x+1,forall x in N`.

Options

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

Explanation

Let `P(n):2.7^(n)+3.5^(n)-5`
Step I For `n=1`,
`P(1):2.7^1+3.5^1-5`
:24 is divisible by 24.
Step II Assume P(k) is divisible by 24, then
`P(k):2.7^k+3.5^k-5=24lambda, lambda` is positive integer.
Step III For `n=k+1`, `P(k+1)-P(k)=(2.7^(k+1)+3.5^(k+1)-5)-(2.7^k+3.5^k-5)`
`=2.7^k(7-1)+3.5^k(5-1)`
`12(7^k+5^k)`
=divisible by 24
`=24 mu , forall mu in I" "[ because 7^k+5^k "is a always divisible by"24]`
`therefore P(k+1)=P(k)+24mu=24lambda+24mu`
`=24(lambda+mu)`
Hence , `P(k=1)` is divisible by 24.
Hence , Statement -1 true and Statement -2 is false .