Question

prove by mathematical induction that 7^n -3^n is divisible by 4.​

Answer 1

We have to prove,

4 | 7^n - 3^n,  n ∈ N

Let n = 1.

7 - 3 = 4   ;   4 | 4

So P(1) holds true.

Let n = k.

Assume  4 | 7^k - 3^k.

Let  7^k - 3^k = 4m,  m ∈ N

⇒   7^k = 4m + 3^k   →   (1)

Let n = k + 1.

    7^(k + 1) - 3^(k + 1)

⇒  7^k × 7  -  3^k × 3

⇒  (4m + 3^k)7 - 3^k × 3   [From (1)]

⇒  4m × 7 + 3^k × 7 - 3^k × 3

⇒  4m × 7 + 3^k(7 - 3)

⇒  4m × 7 + 3^k × 4

⇒  4(7m + 3^k)

⇒  4 | 7^(k + 1) - 3^(k + 1)

Hence Proved!