Prove that 2.7ⁿ + 3.5ⁿ-5 is divisible by 24, for all n belongs to N....Plzzz dnt tell to refer textbook as frm dat also its not clear to me plzzzzzz answer it as soon as possible.....

Let p (n) be the statement given by

∴ p (1) is true

Let p (m) : 2.7^m + 3.5^m – 5 is divisible by 24 be true

⇒ 2.7^m + 3.5^m – 5 = 24λ  ;λ∈N

⇒ 3.5^m = 24λ + 5 – 2.7^m  ...  (1)

Now

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

= 2.7^{m + 1} + (3.5^m) 5  – 5

= 2.7^{m + 1} + (24λ + 5 – 2.7^m) 5  – 5    (from (1))

= 2.7^{m + 1} + 120λ + 25 – 10.7^m  – 5

= (2.7^{m + 1} – 10.7^m) + 120λ + 20

= (2 × 7λ 7^m – 10 × 7^m) + 120λ + 24 – 4

= (14 – 10)7^m – 4 + 24 (5λ + 1)

= 4 (7^m – 1) + 24 (5λ + 1)

= 4 × 6µ + 24 (5λ + 1)  (∵ 7^m – 1 is a multiple of 6 for all m∈N ∴ 7^m – 1 = 6µ, µ∈N)

= 24 (µ + 5λ + 1)  which is divisible by 24

∴ p (m + 1) is true.

Hence by principle of mathematical induction p (n) is true for all n∈N