Sum of n GP Terms: The Compact Formula
S_n = a(r^n - 1)/(r - 1). One line replaces adding ten terms by hand.
Equation
y = 1 * (3^x - 1) / (3 - 1)
Graph
Table
| x | y |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 13 |
| 4 | 40 |
| 5 | 121 |
| 6 | 364 |
| 7 | 1093 |
| 8 | 3280 |
What this lesson covers
What you do
You shape the function y = a * (r^x - 1) / (r - 1) and watch the graph answer.
Challenges to clear
- Set a = 3 and r = 2: the sum of the first 6 terms is 189 — one line instead of six additions.
- S_1 is the first term alone: x = 1 shows 3.
- First term 2 — slide a there.
- Now the ratio: the first 5 terms must total 242. One formula replaces five additions.
Check yourself
Why does the formula S_n = a(r^n - 1)/(r - 1) require r != 1?
- Because at r = 1, the denominator r - 1 is zero (and a GP with r = 1 is just a + a + a + ... = n*a — handle that case separately). — correct
- Because r = 1 makes all terms negative.
- Because the formula only works for whole numbers.
- Because r = 1 gives an undefined sequence.
Think about it
- S_n = a(r^n − 1)/(r − 1) with a = 3, r = 2: what is S_6?