Question

2.7ⁿ + 3.5ⁿ − 5 is divisible by 24 for all n ∈ N.

Answer 1

Given expression

2.7ⁿ + 3.5ⁿ - 5

Proof – Let, P(n) = 2.7ⁿ + 3.5ⁿ - 5 is divisible by 24.

We note that P (n) is true when n = 1,

Since P (1) = 14 + 15 – 5 = 24 which is divisible by 24.

Assume that it is true for P (k).

i.e. P(k) = 2.7^k + 3.5^k - 5 = 24q     ......(1) when q ∈ N

Now according to the mathematical induction principle, we have to prove it is also true for P (k + 1) whenever p (k) is true.

Now substitute in place of k, (k + 1) we have,

⇒ P(k + 1) = 2.7^k+1 + 3.5^k+1 - 5

⇒ 2.7.7^k + 3.5.5^k - 5

Now add and subtract by 3.7.5^k - 7.5 we have,

⇒ 2.7.7^k + 3.5.5^k - 5 + 3.7.5^k - 7.5 - (3.7.5^k - 7.5)

⇒ 7(2.7^k + 3.5^k - 5) + 3.5.5^k - 5 - (3.7.5^k - 7.5)

Now from equation (1) we have,

⇒ 7(24q) + 3.5.5^k - 5 - 3.7.5^k + 7.5

⇒ 7(24q) - 2.3.5^k - 5 + 35

⇒ 7(24q) - 2.3.5^k + 30

⇒ 7(24q) - 6(5^k - 5)

Now as we know that (5^k - 5) is a multiple of 4 so in place of that we can write (4p) where (p) belongs to the natural number.

⇒ 7(24q) - 6(4p)

⇒ 24(7q - p)

So as we see this is a multiple of 24.

Thus P (k + 1) is true whenever P (k) is true.

So according to the principle of mathematical induction 2.7^k + 3.5^k - 5  is divisible by 24.