Question

By using the principal of mathematical induction prove that 5 2n+2 - 24n + k is divisible by 576 for all . then numerically least value of k is___

Answer 1

Prove with explanation:

5^(2n+2) - 24n - 25 is divisivle by 576

If n = 1,

f(n) = 5^4 - 24 - 25 = 625 - 49 = 576.

Assume true for n = r

f(r+1) = 5^(2r + 4) - 24r - 24 - 25

= 25.5^(2r+2) - 24r - 25 - 24

= 25.5^(2r+2) - 25.24r - 25.25 - 24 + 24.24.r + 24.25

= 25.f(r) - 576r + 576

Each term divisible by 576 so whole expression is.

I hope it will help you.