Question

Prove by mathematical induction that (5 raise to 2n)- 1 is divisible by 24.​

Answer 1

Given :

• 5²ⁿ - 1

To Prove :

• (5²ⁿ - 1) is divisible by 24

Proof :

Let us consider,

$\sf P(n) = 5^{2n} - 1$

For n = 1 , we have :

$\sf P(1) = 5^{2\times 1} - 1 \\\\ \sf \implies P(1) = 5^{2} - 1 \\\\ \sf \implies P(n) = 25 - 1 \\\\ \sf \implies P(1) = 24 , \: is \: divisible \: by \: 24$

Now let us assume that

$\sf P(k) = 5^{2k} - 1 \: be \: divisible \: by \: 24 \\\\ \sf\therefore 5^{2k} - 1 = 24q , \: where \: q \: is \: any \: natural \: number.$

For n = k+1 we have :

$\sf P(k+1) = 5^{2(k+1)} - 1 \\\\ \sf \implies P(k + 1) = 5^{2k + 2}-1 \\\\ \sf \implies P(k + 1) = 5^{2k}. 5^{2} -1 \\\\ \sf \implies P(k+1) = 5^{2k} \times 25 -1 \\\\ \sf \implies P(k+1) = 5^{2k} (24 +1 ) - 1 \\\\ \sf \implies P(k +1) = 24 \times 5^{2k} + 5^{2k} - 1 \\\\ \sf\implies P(k+1) = 24 \times 5^{2k} + 24q \\\\ \sf \implies P(k+1) = 24(5^{2k} + q)$

Thus , P(n) is divisible by 24 for n = k + 1 also .

Therefore , by the method of mathematical induction P(n) is divisible by 24 for any natural number n