how to Find the sum of the series 1+2+2^2+2^3+…..+2^n

Find the sum of the series 1+2+2^2+2^3+…..+2^n

Answer :

To Find 1+2+2^2+2^3+…..+2^n We know that any series of the from [a+ar+ar^2+ar^3+\ldots\ldots+ar^{n-1}] is called Geometric Progression. Where a is the first term and r is the common ratio of the progression. Therefore the sum of n terms of a Geometric Progression is obtained by [S_n=\frac{a\left(r^{n+1}-1\right)}{r-1}] For the given series [S_n=1+2+2^2+2^3+\ldots..+2^n] Fo this series first term is a =1 and common ratio r is 2. Hence sum [S_n=\frac{a\left(r^{n+1}-1\right)}{r-1}=2^{n+1}-1] Therefore [1+2+2^2+2^3+\ldots..+2^n=2^{n+1}-1]