Geometric Series
In short: A geometric series is the sum of terms in which each one is a fixed multiple of the one before it, and its total has a closed formula instead of requiring term-by-term addition.
Algebra 2AlgebraA-SSE.B.4
What is Geometric Series?
A geometric series adds numbers that grow, or shrink, by multiplication: each term is the previous term times a constant ratio \(r\). Chain letters, repeated discounts and bouncing balls behave this way.
With first term \(a\) and ratio \(r \neq 1\), the sum of the first \(n\) terms is \(S_n = \frac{a(1 – r^n)}{1 – r}\). If \(|r| < 1\), the terms fade away and the infinite sum exists: \(S = \frac{a}{1 - r}\). When \(|r| \ge 1\), the infinite sum does not settle on a number.
How to spot it
- Each term in the list is found by multiplying the previous one by the same number, so a geometric series is being added.
- A story involves doubling, tripling or repeated percentage change, and the total over many rounds is wanted.
- A repeating decimal or a bouncing ball is being summed over infinitely many steps.
- The notation sum of a·r^(k-1) appears, with r a constant.
The method
- Find the first term \(a\) and divide any term by the one before it to get the ratio \(r\).
- For \(n\) terms, use \(S_n = \frac{a(1 – r^n)}{1 – r}\).
- For an infinite geometric series, first confirm \(|r| < 1\), then use \(S = \frac{a}{1 - r}\).
- Compute with care for powers and negative ratios, and round only at the end.
Geometric Series: solved example
Problem
A chain message starts with 6 people. Each round, every new group sends it to three times as many people as the round before: 6, 18, 54, and so on. How many messages go out in 7 rounds?
Solution
The first term is \(a = 6\) and the ratio is \(r = 3\), with \(n = 7\). Then \(S_7 = \frac{6(1 – 3^7)}{1 – 3}\).
Since \(3^7 = 2187\), \(S_7 = \frac{6(-2186)}{-2} = 6 \times 1093 = 6{,}558\). A total of 6,558 messages are sent.
Try it
Question 1
Find the infinite sum \(18 + 6 + 2 + \cdots\).
Show the answer
The ratio is \(\frac{1}{3}\), so \(S = \frac{18}{1 – \frac{1}{3}} = 27\).
Question 2
Find \(S_8\) for \(a = 5\) and \(r = 2\).
Show the answer
\(S_8 = \frac{5(1 – 2^8)}{1 – 2} = 5 \times 255 = 1{,}275\).
Geometric Series vs. Arithmetic Series
Multiplication drives this one: each term is the previous term times \(r\). In the arithmetic version each term adds a constant. Growth also differs: adding 3 each step creates straight-line growth, while multiplying by 3 each step creates exponential growth that explodes after only a few terms.
Keep going
Lessons
Worksheets and quizzes
- Arithmetic and Geometric Sequences and Series – Worksheet
- Arithmetic and Geometric Sequences and Series – Quiz
- Sequences and Series – Worksheet
- Sequences and Series – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Algebra