Geometric Series (Summing a GP)
Learn geometric series for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). Adding the terms of a geometric progression by hand quickly becomes slow, but one formula gives the sum of \(n\) terms whenever the common ratio is not 1.
You will learn to apply the sum formula for a GP, choose the version that suits a ratio above or below 1, find how many terms reach a target total, and handle problems on compound growth and repeated payments — a key tool for the financial mathematics later in the course.
Theory
A geometric series is the sum of the terms of a geometric sequence. This Year 12 Mathematics Advanced topic (MAV-12-03) uses \(S_n=\dfrac{a(r^{\,n}-1)}{r-1}\) (for \(r\neq1\)) to find sums.
The sum of the first \(n\) terms of a GP is \(S_n=\dfrac{a(r^{\,n}-1)}{r-1}=\dfrac{a(1-r^{\,n})}{1-r}\), valid whenever \(r\neq1\).
The two forms are identical — use the first when \(r>1\) and the second when \(0
Method
- Identify \(a\), \(r\) and \(n\) (find \(n\) from the last term if needed).
- Choose the form: \(r>1\) use \((r^n-1)/(r-1)\); \(0
- Substitute and evaluate.
| \(S_6\) | \(=\) | \(\dfrac{2(3^6-1)}{3-1}\) |
| \(=\) | \(728\) |
| \(n\) | \(=\) | \(8\ (128=2^7)\) |
| \(S_8\) | \(=\) | \(\dfrac{2^8-1}{1}=255\) |
| \(S_5\) | \(=\) | \(\dfrac{100\big(1-(\tfrac12)^5\big)}{1-\tfrac12}\) |
| \(=\) | \(193.75\) |
| \(r\) | \(=\) | \(3\) |
| \(S_4\) | \(=\) | \(\dfrac{5(3^4-1)}{2}=200\) |
Common pitfalls
Frequently asked questions
What is the formula for the sum of a geometric series?
It is S n equals a times (r to the n minus 1) divided by (r minus 1), for r not equal to 1, where a is the first term and r the common ratio.
When do you use 1 minus r instead of r minus 1?
They give the same answer. Use a times (1 minus r to the n) over (1 minus r) when r is between 0 and 1, to keep the numbers positive.
How do you find the number of terms in a geometric series?
Use the nth term formula a n equals a times r to the power n minus 1, set it to the last term, and solve for n.
What is the difference between a geometric series and its limiting sum?
The series formula adds a fixed number of terms. The limiting sum adds infinitely many terms and only exists when the ratio is between minus 1 and 1.