# Daniel’s Sum of Geometric Series

Continuing with the search of young Mathematicians, Daniel James Agsaullo Molina from Saint Louis College, La Union wants to share one of his favorite problems. He is an incoming freshman student. He was one of the national finalists in the individual category of 2014 Metrobank MTAP-Dep-Ed Math Challenge representing region 1.

**Problem:**

Given a geometric sequence whose sum of the first 20 terms is 7. And whose sum from the 21st to the 60^{th} term is 84, find the sum from the 61^{st} to the 120^{th} term.

**Solution:**

The sum of a geometric sequence is defined as:

Where n=number of terms, a=first term, and r = common ratio

Expressing the sum of the first 20 terms,

…#1

Another given is the sum of the terms from the 21^{st} to the 60^{th} which is 84.

It is expressed as

Therefore,

Expressing the sum ,

We get

…#2

From #2 ÷ #1 and simplifying,

Since as r ranges to all possible common ratio.

…#3

We are ask to find the sum of 61^{st} to 120^{th}

***main equation

Where

But and from #3,

Substituting to our main equation,