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Login NowLet, p(n) = 5^{n} – 1

**Basis Step:**

We show, p(1) is true

i.e. p(1) = 5^{1} – 1

= 4 which is divisible by 4

so p(1) is true

**Inductive Step:**

Suppose for any arbitary value k, p(k) is true

i.e. p(k): 5^{k} – 1 = 4a, where, a is an integer

or, 5^{k} = 4a + 1

Now, we need to show p(k+1) is divided by 4

p(k+1) = 5^{k+1} – 1

= 5^{k}.5 – 1

= (4a+1).5 -1

= 20a + 5 – 1

= 20a + 4

= 4(5a + 1)

Here, it is divisible by 4

So, by mathematical induction

5^{n-1} is divisible by 4

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