Geometric Sequence and Series

A geometric sequence multiplies by the same ratio each step. A geometric series adds the terms of that sequence. If the first term is $a_1$ and the common ratio is $r$, then the sequence formula is $a_n = a_1r^{n-1}$, and the finite sum formula is $S_n = \frac{a_1(1-r^n)}{1-r}$ when $r \ne 1$.

For example, $3, 6, 12, 24$ is geometric because each term is found by multiplying by $2$. Use the sequence formula when you want a term. Use the series formula when you want the total of several terms.

What makes a sequence geometric

The key idea is a constant ratio. In an arithmetic sequence, you add the same amount each time. In a geometric sequence, you multiply by the same amount each time.

If the first term is $a_1$ and the ratio is $r$, then

$$a_n = a_1r^{n-1}$$

If $r$ is negative, the signs alternate. If the absolute value of $r$ is less than $1$, the terms get smaller in size.

Geometric sequence vs. geometric series

A geometric sequence is the list of terms. A geometric series is the sum of those terms.

That difference matters because the question changes what you should compute. "Find the fifth term" asks for a sequence value. "Find the sum of the first five terms" asks for a series value.

Worked Example: Find a Term and a Finite Sum

Use the geometric sequence

$$3,\ 6,\ 12,\ 24,\ 48$$

Here, $a_1 = 3$ and $r = 2$.

To find the fifth term:

$$a_5 = 3 \cdot 2^{5-1} = 3 \cdot 16 = 48$$

To find the sum of the first five terms, add the terms directly:

$$S_5 = 3 + 6 + 12 + 24 + 48 = 93$$

You can also use the finite geometric series formula:

$$S_n = \frac{a_1(1-r^n)}{1-r}$$

For this example,

$$S_5 = \frac{3(1-2^5)}{1-2} = 93$$

When the Geometric Series Formula Works

For a finite geometric series, the formula

$$S_n = \frac{a_1(1-r^n)}{1-r}$$

works when $r \ne 1$.

If $r = 1$, every term is the same, so the sum is just

$$S_n = na_1$$

For an infinite geometric series, there is a finite sum only when the absolute value of $r$ is less than $1$.

Common Mistakes

1. Using a common difference instead of a common ratio.
2. Mixing up a term question with a sum question.
3. Using the finite sum formula when $r = 1$, which would divide by zero.
4. Forgetting that a negative ratio makes the signs alternate.

When Geometric Sequences and Series Are Used

Geometric patterns appear when change happens by a constant factor. That includes doubling, repeated percentage decay, compound growth, and some infinite-series ideas in calculus.

Try Your Own Version

Try a new sequence with first term $5$ and common ratio $\frac{1}{2}$. Find the first four terms, then find their sum. If you want another case, try a negative ratio and check how the signs change from term to term.