Prove that 5n 4n 1 is divisible by 16 for all positive int

Prove that 5^n - 4n - 1 is divisible by 16 for all positive integer n.

Solution:

Let p(n) = 5ⁿ - 4n - 1 which is divisible by 16

We prove by mathematical induction method.

case 1): put n = 1, we get p(1) = 5¹ - 4(1) - 1 = 0 which is divisible by 16

case 2): put n=k, we get p(k) = 5^k - 4k - 1 . By induction hypothesis p(k) is also divisible by 16

case 3): put n = k+1, we get p(k+1) = 5^k+1 - 4(k+1) - 1

= 5^k+1 - 4k - 5

= (5^k+1 - 20k - 5) + 16k

= 5(5^k - 4k - 1) + 16k

= 5p(k) + 16k

Since, p(k) is divisible by 16, therefore, p(k+1) is also divisible by 16.

Thus, by mathematical induction p(n) is divisible by 16.

Hence proved.