Math, asked by Anonymous, 1 month ago

5^n+2-5^n/24×5n^n=1 prove​

Answers

Answered by ITZURADITYATYAKING
1

Answer:

Let 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 have 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

⇒  2×24q−21.5k+35+15.5k−5

⇒  7×24q−6.5k+30

⇒  7×24q−6(5k−5)

⇒  7×24q−6(4p)                   [ (5k−5) is multiple of 4 ]

⇒  7×24q−24p

⇒  24(7p−q)

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

The expression on the R.H.S oof ( 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.   

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