Sum of a Finite Geometric Series

A geometric series adds terms that each multiply by a fixed ratio $r$ , like $3 + 6 + 12 + \cdots$ . Adding many terms by hand is slow, so there is a formula.

The sum of the first $n$ terms is $S_n = a\dfrac{1 - r^n}{1 - r}$ , where $a$ is the first term. Identify $a$ , $r$ , and $n$ , then plug in.

Identify the three pieces

The first term is $a$ . The common ratio $r$ is any term divided by the one before it. For $3 + 6 + 12$ , $\displaystyle r = \frac{6}{3} = 2$ .

$n$ is how many terms you are adding. With those three numbers the formula does the rest.

Applying the formula

Substitute into $S_n = a\dfrac{1 - r^n}{1 - r}$ and simplify the power $r^n$ carefully.

The formula works for any ratio except $r = 1$ (where every term is the same and you just multiply $a$ by $n$ ).

Worked examples

Example 1: sum of five terms

Find the sum of the first 5 terms of $3 + 6 + 12 + \cdots$

Identify a, r, n

a = 3, ; r = 2, ; n = 5

Apply the formula

3 \cdot \dfrac{1 - 2^5}{1 - 2}

Simplify

3 \cdot \dfrac{-31}{-1} = 93

Answer: $93$

Example 2: finding the ratio

What is the common ratio of $5 + 10 + 20 + \cdots$ ?

Divide a term by the previous

10 \div 5

Simplify

2

Answer: $r = 2$

Example 3: when the ratio is 1

Find the sum of the first 6 terms of $7 + 7 + 7 + \cdots$

Divide a term by the previous

r = 7 \div 7 = 1

The formula's denominator would be zero

1 - r = 0

Every term equals a, so multiply by n

7 \cdot 6

Simplify

42

Answer: $42$

Try one yourself

Common questions

How do I find the common ratio?

Divide any term by the term before it. If that ratio is constant, the series is geometric and that value is $r$ .

What are a, r, and n?

$a$ is the first term, $r$ the common ratio, and $n$ the number of terms being added.

Does the formula work if r = 1?

No — it would divide by zero. If $r = 1$ every term equals $a$ , so the sum is simply $a \cdot n$ .

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Sum of a Finite Geometric Series

Identify the three pieces

Applying the formula

Worked examples

Example 1: sum of five terms

Example 2: finding the ratio

Example 3: when the ratio is 1

Try one yourself

Common questions

Keep learning

Geometric Sequences

Graphing Exponential Functions

Modeling Data: Choosing a Model & Regression