Let the given statement is P(n), where n ∈ N,
When n = 1, then
P(1) = 1/2 = 1 - 1/21 = 1 - 1/2 = 1/2
Hence, given statement is true for n = 1,
i.e, P(1) is true
P(m) = 1/2 + 1/4 + 1/8 + .. + 1/2m = 1 - 1/2m ......(i)
Now, we have to prove that given statement is true for n = m + 1
i.e, P(m + 1) is also true.
Now,
Hence, the given statement is also true for n = m + 1,
i.e., P(m + 1), is true.
Hence, by the principle of mathematical induction, we can say that the given statement is true for each natural number n ∈ N.
Hence Proved.