# Geometric Progression - Sum it upto n-terms

• Jun 23rd 2010, 11:20 PM
sureshrju
Geometric Progression - Sum it upto n-terms
Sum the series upto n-terms : 1+(1+x)+(1+x+x^2)+(1+x+x^2+x^3)...............

Answer in the book is [n/(1-x)] - [x(1-x)^n / (1-x)^2]

The answer is in the book, but i dont know how to find it. I need the full solution for the problem by step by step. Can you please anybody give me help to solve this?
• Jun 24th 2010, 12:05 AM
red_dog
The sum can be written as
$\displaystyle\sum_{k=0}^{n-1}(1+x+x^2\ldots+x^k)=\displaystyle\sum_{k=0}^{n-1}\frac{1-x^{k+1}}{1-x}=$

$=\displaystyle\sum_{k=0}^{n-1}\frac{1}{1-x}-\displaystyle\sum_{k=0}^{n-1}\frac{x^{k+1}}{1-x}=\frac{n}{1-x}-\frac{x(1-x^{n})}{(1-x)^2}$