How is (5 ^{k _}5) a multiple of 4 ?

Let P(k) be the statement given by 5^k – 5.

⇒ P(k) = 5^k – 5

For k = 1, P (1) = 5¹– 5 = 0

For k = 2, P (2) = 5²– 5 = 25 – 5 = 20 which is a multiple of 4.

Let P(k) be true i.e., P(k)  is a multiple of 4.

⇒ P(n) = 5^k – 5  is a multiple of 4

⇒ 5 [ 5^k–1 – 1] = 4t, for some integer t. ....... (1)

We will now prove that P(k + 1) is true.

⇒ P(k + 1) = 5^k+1 – 5 = 5 [ 5^k – 1 ]

Hence, by the principle of mathematical indication P(k) is true for all k > 1.