So Here's a few problems where I have to find the nth partial sum:

Given the equation

The sum of a Geometric series for a sequence such that it follows the general pattern starting at n=1. a is a constant, and

|r| < 1

If we let n go towards infinity, then

However, before I can begin dealing with infinity, I can't get the general equation for the sum up to the nth term. What am I doing wrong?

The Sequence is:

if we let a = 10 and r = 1/5, then from Sn I got

That's what I got. However the way it is done is:

let a = 2 and let r= 1/5

Manual calculation by hand shows the second method to be the correct one. While I accept it, I can't see what's wrong with the first.