Question

For all n ≥ 1 , prove that p(n): 2.7ⁿ + 3.5ⁿ – 5 is divisible by 24.

Answer 1

p(1): 2.7¹ + 3.5¹ – 5 = 14 + 15 – 5 = 24 divisible by 24, hence true. Assuming that true for p(k)

p(k): 2.7^k + 3.5^k – 5 is divisible by 24.

⇒ 2.7^k + 3.5^k – 5 = 24M

p(k + 1): 2.7^{k + 1} + 3.5^{k + 1} – 5

= 2.7^k.7 + 3.5^k.5 – 5

= 2.7^k.(6 + 1) + 3.5^k.(4 + 1) – 5

= 12.7^k + 2.7^k + 12.5^k + 3.5^k – 5

= 12(7^k + 5^k) + (2.7^k + 3.5^k) – 5

= 12(7^k + 5^k) + 24M

7^k And 5^k are odd numbers, therefore (7^k + 5^k) will be an even and will be divisible by 24, Hence p(k + 1)divisible by 24. 

Therefore by using the principle of mathematical induction true for all n ∈ N.