Q. 17

# Prove the following by the principle of mathematical induction:

5^{2n + 2} – 24n – 25 is divisible by 576 for all n ϵ N.

Answer :

Let P(n): 5^{2n + 2} – 24n – 25

For n = 1

= 5^{2.1 + 2} - 24.1 - 25

= 625 – 49

= 576

Since, it is divisible by 576

Let P(n) is true for n=k, so

= 5^{2k + 2} – 24k – 25 is divisible by 576

= 5^{2k + 2} – 24k – 25 = 576λ - - - - - (1)

We have to show that,

= 5^{2k + 4} – 24(k + 1) – 25 is divisible by 576

= 5^{(2k + 2) + 2} – 24(k + 1) – 25 = 576μ

Now,

= 5^{(2k + 2) + 2} – 24(k + 1) – 25

= 5^{(2k + 2)}.5^{2} – 24k – 24– 25

= (576λ + 24k + 25)25 – 24k– 49 using equation (1)

= 25. 576λ + 576k + 576

= 576(25λ + k + 1)

= 576μ

Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n∈N by PMI

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