Question

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solve the question given in attachment...

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Answer 1

P ( n ) = 2 . 7ⁿ + 3 . 5ⁿ - 5

p ( 1 ) = 2 . 7¹ + 3 . 5¹ - 5 = 14 + 15 - 5 = 24 , which is divisible by 24

Therefore p ( 1 ) is true !

Assume that p ( k ) is true .

$P ( k ) = 2 . 7 {}^{k} + 3 . 5 {}^{k} - 5 = 24p \: \: \: .....(1)$
where p is any natural number

We have to prove that p ( k + 1 ) is true

$P ( k \: + 1 ) = 2 . 7 {}^{k + 1} + 3 . 5 {}^{k + 1} - 5 \\ = 2 . 7 {}^{k} \times 7 + 3 . 5 {}^{k} \times 5 - 5 \\ = 7(2. {7}^{k} + 3.5 {}^{k} - 5 - 3.5 {}^{k} + 5) + 3.5 {}^{k} \times 5 - 5 \\ = (7 \times (24p) - 21.5 {}^{k} + 35 )+ 3.5 {}^{k} \times 5 - 5 \: \: .......see \: (1) \\ = 7(24p) - 6.5 {}^{k} + 30 \\ = 7(24p) - 6(5 {}^{k} - 5) \\ = 7(24p) - 6(4q) \\ = 24(7p - q) \\ \\ therefore \: it \: is \: divisible \: by \: 24$

Hence proved

Here ,

➡ 2.7 refers to 2 × 7 , like that 3.5 too

⏩ How 5ⁿ - 5 became 4q ?

Answer : Substitute any values for n !

When n = 2 , 5ⁿ - 5 = 25 - 5 = 20 = 4 × 5

when n = 3 , 5ⁿ - 5 = 125 - 5 = 120 = 4 × 30

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Like that when n = k , 5ⁿ - 5 = 4 × q ,where q is any natural number and k is any natural number