Question

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

Answer 1

Let P(n) ≡ 5²ⁿ – 2²ⁿ is divisible by 3, for all n ∈ N.

Step 1:

For n = 1, 5²ⁿ – 2²ⁿ = 5² – 2² = 25 – 4 = 21, which is divisible by 3.

∴ P(1) is true.

Step 2:

Let us assume that for some k ∈ N, P(k) is true, i.e. 5^2k – 2^2k is divisible by 3.

∴ ${\displaystyle \frac{5^{2\text{k}}-2^{2\text{k}}}{3}}$ = m (Say), whre m ∈ N

∴ 5^2k –  2^2k = 3m

∴ 5^2k = 2^2k + 3m   ...(1)

Step 3:

To prove that P(k + 1) is true, i.e., to prove that ${\displaystyle 5^{2(\text{k}+1)}-2^{2(\text{k}+1)}}$ is divisible by 3.

Now, ${\displaystyle 5^{2(\text{k}+1)}-2^{2(\text{k}+1)}}$ = 5^2k+2 – 2^2k+2 

= 5^2k .5² – 2^2k . 2²

= (2^2k + 3m)25 – 2^2k . 4    ...[By (1)]

= 25(2^2k) + 75m – 4(2^2k)

= 21(2^2k) + 75m

= 3[7.2^2k + 25m]

∴ ${\displaystyle \frac{5^{2(\text{k}+1)}-2^{2(\text{k}+1)}}{3}}$ = 7.2^2k + 25m, where (7.2^2k + 25m) ∈ N

∴ ${\displaystyle 5^{2(\text{k}+1)}-2^{2(\text{k}+1)}}$ is divisible by 3

∴ P(k + 1) is true.

Step 4:

From all the above steps and by the principle of mathematical induction P(n) is true for all n ∈ N,

i.e., 5²ⁿ – 2²ⁿ is divisible by 3, for all n ∈ N.