Question

provoke that for all positive integer n 7^n-3^n is divisible by 4

Answer 1

let the given statement be P(n) : 7^n - 3^n is divisible by 4.
put n = 1
= 7^1 - 3^1
= 7 - 3
= 4
Hence, p(1) is true.
let p(k) be true.
P(k) : 7^k - 3^k = 4m
Now,
7^k+1 - 3^k+1
= 7^k x 7 - 3^k x 3
on subtracting and adding 7 x 3^k
=7^k x 7 - 7 x 3^k + 7 x 3^k - 3^k x 3
= 7(7^k - 3^k) + 3^k(7 -3)
= 7 x 4m + 4 x 3^k
= 4(7^k - 3^k), which is divisible by 4.