Let the given statement be P(n), i.e.,
P(n):12+14+18+.....+12n=1−12n
For n=1, we have
P(1)=12=1−121=12, which is true.
Let P(k) be true for some k∈N, i.e.,
12+14+18+.....+12k=1−12k......(i)
We shall now prove that P(k+1) is true.
Consider (12+14+18+.....+12k)+12k+1
=(1−12k)+12k+1
=1−12k+12.2k
=1−12k(1−12)
=1−12k(12)
=1−12k+1
Thus P(k+1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n