Question

Given that 5ⁿ-5 is divisible by 4 ⊂ N. Prove that 2.7ⁿ + 3.5ⁿ−5 is multiple of 24.

Answer 1

Let P(n) ∶ 2.7ⁿ + 3.5ⁿ − 5 is a multiple of 24.

Step 1: P(1) ∶ 2.7ⁿ + 3.5¹ − 5 = 14 + 15 − 5 = 24,

Which is a multiple of 24.

∴ P(1) is true.

Step 2: Let it be true for some k ∈ N, i.e,.

P(k): 2.7k + 3.5k − 5 = 24λ for some λ ∈ N.  ...(i)

Step 3: To prove that

P(k + 1) ∶ 2.7^k+1+3.5^k+1−5 is also multiple of 24.

Now, 

2.7^k+1+3.5^k+1−5

2.7^k.7 + 3.5^k.5 − 5

7(2.7^k) + 5.(3.5^k) − 5

7(24λ− 3.5^k + 5) + 5.(3.5^k) − 5

= 7.24λ − 21.5^k + 15.5^k + 35 − 5

= 7.24λ − 6(5^k−5)

= 7.24λ − 6(4μ)

= 24(7λ − μ)

Which is a multiple of 24. For some λ,μ ∈ N.

∴ P(k + 1) is true whenever P(k) is true

Hence, by Principle of Mathematical Induction

P(n) in true for all n ∈ N.