Prove that 2.7^{n} + 3.5^{n}-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

*– 5 is divisible by 24 be true*

^{m}⇒ 2.7* ^{m}* + 3.5

*– 5 = 24λ ;λ∈N*

^{m}⇒ 3.5* ^{m }*= 24λ + 5 – 2.7

*... (1)*

^{m}

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

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