1/2+1/4+1/8+..........+1/2ⁿ=(1-1/2ⁿ)

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Solution

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 positive integer , 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 [Using (i)]
=1−12k+12.2k
=1−12k(1−12)
=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.

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