Question

Heyaa !!!

Prove that
${2.7}^{n} \: + \: {3.5}^{n} - \: 5$
is divisible by 24 for all n € N



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Answer 1

Hiii bcha

Step-by-step explanation is in attachment✌

#aashiq_g❤

@skb♥️

Answer 2

Hiii pp ji,☺️☺️

Let the statement P(n) be defined as

P(n):2.7n+3.5n−5 is divisible by 24

We note that P(n) is true when n=1, since 2.7+3.5−5=24, which is divisible by 24.

Assume that P(k) is true.

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

Now, we wish to prove that P(k+1) is true whenever P(k) is true.

We have

2.7k+1+3.5k+1−5=2.7k.71+3.5k.51−5

=7[2.7k+3.5k−5−3.5k+5]+3.5k.5−5

=7[24q−3.5k+5]+15.5k−5

=7×24q−21.5k+35+15.5k−5

=7×24q−6.5k+30

=7×24q−6(5k−5)

=7×24q−6(4p)[(5k−5) is a multiple of 4 (why ?) ]✋✋

=7×24q−24p

=24(7q−p)

=24×r;r=7q−p, is some natural number-----------(2)

The expression on the R.H.S of (1) is divisible by 24. Thus P(k+1) is true whenever P(k) is true.

Hence, by principle of mathematical induction , P(n) is true for all n∈N.✌️✌️

I hope aapko ye smjh aa gya hoga☺️☺️❤️❤️