Prove the following by the principle of mathematical induction:
Let P(n) : 1 + 3 + 32 + - - - - + 3n - 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 + - - - - + 3k - 1 = - - - - - (1)
Now, We have to show P(n) is true for n = k + 1
i.e P(k + 1): 1 + 3 + 32 + - - - - + 3k =
then, {1 + 3 + 32 + - - - - + 3k - 1} + 3k + 1 - 1
= using equation (1)
=
=
=
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n∈N