1+3+3² +… +3^n-1 = .

Let P(n) = 1+3+3² +… +3^n-1
P(1) is true
Assuming P(k) is true
1+3+3² +… +3^k-1 =
To prove P(k +1) is true using the result of P(k)
1+3+3² +… +3^k-1 +3k = ……………(1)
proof: Taking L.H.S of (1)
1+3+3² +… +3^k-1 +3k
                               = + 3k
                               = =
                               =

Which is the R.H.S of (1)

P(K + 1 ) is true

By the Principle of Mathematical induction, given expression is true for all values of n .
When n N
Hence proved.