Thread: Induction Proof

Prove that (5^2n) - 1 is a multiple of 8 for all natural numbers. (By induction)

Originally Posted by bearej50

Prove that (5^2n) - 1 is a multiple of 8 for all natural numbers. (By induction)

.....Here is the main part
Let 5^2k-1 is a multiple of 8
so
5^{2k}-1 =8 m [ m is any integer]

5^2k = 8m+1

For(k+1)

$\displaystyle 5^{2(k+1)}-1$

$\displaystyle =25*5^{2k}-1$

$\displaystyle
=25*(8m+1)-1
$
$\displaystyle =25*8m+24$

$\displaystyle =8(25m+3)$

thank you!