Algebra 2
Sequences and series
Finite geometric series formula
20 practice questions
0 video lessons
Theory + worked examples
Theory
The finite geometric series sum has a closed formula:
\[S_n=\dfrac{a_1(1-r^n)}{1-r},\qquad r\neq1.\]
It comes from multiplying \(S_n\) by \(r\) and subtracting, which cancels the middle terms.
If \(r=1\) the formula breaks (division by \(0\)); the sum is just \(n\,a_1\).
The finite geometric sum formula.
The multiply-and-subtract derivation.
The sum of \(n\) terms:
\[S_n=\dfrac{a_1(1-r^n)}{1-r}\]
Watch the signs when \(r>1\): both parts are negative.
How to use the formula
- Identify \(a_1\), \(r\), and \(n\).
- Compute \(r^n\).
- Substitute into \(S_n=\dfrac{a_1(1-r^n)}{1-r}\).
- Simplify carefully with signs.
Example 1 — Apply the formula
Find \(3+6+12+24+48\).
Solution
Geometric: \(a_1=3,\ r=2,\ n=5\).
| \(S_5\) | \(=\) | \(\dfrac{3(1-2^5)}{1-2}\) |
| \(=\) | \(\dfrac{3(-31)}{-1}=93\) |
Example 2 — Ratio 3
Find \(1+3+9+27\).
Solution
\(a_1=1,\ r=3,\ n=4\).
| \(S_4\) | \(=\) | \(\dfrac{1(1-3^4)}{1-3}\) |
| \(=\) | \(\dfrac{-80}{-2}=40\) |
Example 3 — A fractional ratio
Find \(16+8+4+2\).
Solution
\(a_1=16,\ r=\dfrac12,\ n=4\).
| \(S_4\) | \(=\) | \(\dfrac{16\left(1-\dfrac1{16}\right)}{1-\dfrac12}\) |
| \(=\) | \(\dfrac{15}{\dfrac12}=30\) |
Example 4 — When r = 1
What is the sum if \(r=1\)?
Solution
Every term equals \(a_1\), so the formula is not needed.
| \(S_n\) | \(=\) | \(n\cdot a_1\) |
Common pitfalls
Count \(n\) correctly — it is the number of terms.
\(r=1\) needs \(S_n=n a_1\) instead of the formula.
Signs: for \(r>1\), numerator and denominator are both negative.
Frequently asked questions
What is the finite geometric series formula?
\(S_n=\dfrac{a_1(1-r^n)}{1-r}\).
How is the formula derived?
Multiply the sum by \(r\), subtract, and solve — the middle terms cancel.
What if \(r=1\)?
All terms are equal, so the sum is \(n\,a_1\).
What is \(n\) in the formula?
The number of terms being added.
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