Prove the following by the principle of mathematical induction:

Let P(n) : 1 + 3 + 3² + - - - - + 3^{n - 1} =

Now, For n =1

P(1): 1 = =1

Therefore, P(n) is true for n =1

Now , P(n) is true for n = k

P(k) : 1 + 3 + 3² + - - - - + 3^{k - 1} = - - - - - (1)

Now, We have to show P(n) is true for n = k + 1

i.e P(k + 1): 1 + 3 + 3² + - - - - + 3^k =

then, {1 + 3 + 3² + - - - - + 3^{k - 1}} + 3^{k + 1 - 1}

= using equation (1)

=

=

=

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

Hence, P(n) is true for all n∈N