Question

What is the sum of the geometric series: 3∑5(-2/3)^ii=0

Answer 1

We use the law of the sum of the 3 first terms of the geometric sequence:
$\sum_{i=0}^35\left(\dfrac{-2}{3}\right)^i=5\dfrac{1-\left(\dfrac{-2}{3}\right)^4}{1-\left(\dfrac{-2}{3}\right)}\\=5\dfrac{\left(\dfrac{3^4-2^4}{3^4}\right)}{\left(\dfrac{5}{3}\right)}\\=3\left(\dfrac{65}{81}\right)=\dfrac{65}{27}$