Prove by the principle of mathematical induction:52n + 2 – 24n –

1.	Prove by the principle of mathematical induction:52n + 2 – 24n – 25 is divisible by 576 for all n ϵ N
Answer» Suppose P (n): 5^{2n + 2} – 24n – 25 is divisible by 576 Now let us check for n = 1, P (1): 5^{2.1 + 2} – 24.1 – 25 : 625 – 49 : 576 P (n) is true for n = 1. Where, P (n) is divisible by 576 Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true. P (k): 5^{2k + 2} – 24k – 25 is divisible by 576 : 5^{2k + 2} – 24k – 25 = 576λ …. (i) Now we have to prove, 5^{2k + 4} – 24(k + 1) – 25 is divisible by 576 5^{(2k + 2) + 2} – 24(k + 1) – 25 = 576μ Therefore, = 5^{(2k + 2) + 2} – 24(k + 1) – 25 = 5^{(2k + 2)}.5² – 24k – 24 – 25 = (576λ + 24k + 25)25 – 24k– 49 by using equation (i) = 25. 576λ + 576k + 576 = 576(25λ + k + 1) = 576μ P (n) is true for n = k + 1 Thus, P (n) is true for all n ∈ N.

Prove by the principle of mathematical induction:52n + 2 – 24n – 25 is divisible by 576 for all n ϵ N

Answer»

Suppose P (n): 5^{2n + 2} – 24n – 25 is divisible by 576

Now let us check for n = 1,

P (1): 5^{2.1 + 2} – 24.1 – 25

: 625 – 49

: 576

P (n) is true for n = 1. Where, P (n) is divisible by 576

Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.

P (k): 5^{2k + 2} – 24k – 25 is divisible by 576

: 5^{2k + 2} – 24k – 25 = 576λ …. (i)

Now we have to prove,

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

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

Therefore,

= 5^{(2k + 2) + 2} – 24(k + 1) – 25

= 5^{(2k + 2)}.5² – 24k – 24 – 25

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

= 25. 576λ + 576k + 576

= 576(25λ + k + 1)

= 576μ

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

Thus, P (n) is true for all n ∈ N.