Prove that 4 divides 5n 1 for all n NSolutionwe can prove i

Prove that 4 divides 5^n - 1 for all n N.

Solution

we can prove it my the method of finite mathematical induction.

Let S(n) be the statement i.e., S(n) = 5ⁿ-1 is divisible by 4

Step:-1 Prove S(n) is true for n= 1.

S(1) = 5 - 1 = 4

Therefore it is divided by 4

Step:-2 Let S(n) is true for n = k

S(k) = 5^k-1 = 4m

5^k = 4m + 1 ------>1

Step:-3 Prove that S(n) is true for n = k + 1

now,

S(k+1) = 5^k+1 - 1

=5^k.5 - 1

from 1

=(4m+1).5 - 1

=20m+5-1

=20m + 4

=4.(5m+1)

=4Q Where Q = 5m + 1

Therefore, it is divisible by 4

Therefore, S(n) is true for n = k+1

By the method of finite mathematical induction S(n) is true for all n belongs to N