Algebra 2 / Precalculus · Step-by-step guide

Geometric Sequences

Find terms and sums of geometric sequences using common ratios and exponential structure.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A geometric sequence multiplies by the same common ratio r each time. Its nth term is an=a1rn−1\displaystyle a_n=a_1r^{n-1}, and a finite sum is Sn=a1(1−rn)/(1−r)\displaystyle S_n=a_1(1-r^n)/(1-r) for r ≠ 1.

an=a1rn−1\displaystyle a_n=a_1r^{n-1}
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Look for equal ratios

Divide a term by the preceding term. A constant result identifies a geometric sequence.

2

Exponent counts multiplications

Reaching term n from term 1 requires n − 1 multiplications by r.

3

Track signs

A negative ratio creates alternating signs; a ratio between −1 and 1 makes magnitudes shrink.

02

See the structure

A picture makes the relationships easier to remember.

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Each term doubles: 1, 2, 4, 8, 16, 32. The ratio stays 2 while the differences grow.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Find a term

Find the 8th term of 3, 6, 12, …

  1. a1=3\displaystyle a_1=3 and r=2\displaystyle r=2.
  2. a8=3(2)7\displaystyle a_8=3(2)^7.
  3. There are seven multiplications.
Answer384
Example 2Negative ratio

Find a formula for 10, −5, 2.5, …

  1. r=−1/2\displaystyle r=-1/2.
  2. Use exponent n − 1.
  3. Keep the ratio in parentheses.
Answeran=10(−1/2)n−1\displaystyle a_n=10(-1/2)^{n-1}
Example 3Finite sum

Sum the first 5 terms of 2, 6, 18, …

  1. a1=2,r=3,n=5\displaystyle a_1=2,r=3,n=5.
  2. Use the finite geometric sum formula.
  3. S5=2(1−35)/(1−3)\displaystyle S_5=2(1-3^5)/(1-3).
Answer242
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Subtracting terms instead of dividing to find r.

  • Using exponent n rather than n − 1.

  • Dropping parentheses around a negative ratio.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Sequences and Series

Compare arithmetic and geometric sequences and series.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find the 6th term of 3, 6, 12, …

Show hint

Use ratio 2.

Show worked solution
  1. a1=3,r=2\displaystyle a_1=3,r=2.
  2. a6=3(2)5=96\displaystyle a_6=3(2)^5=96.

Answer: 96\displaystyle 96

2

Find r for 81, 27, 9, …

Show hint

Divide a term by the previous term.

Show worked solution
  1. r=27/81\displaystyle r=27/81.
  2. Simplify to 1/3\displaystyle 1/3.

Answer: r=1/3\displaystyle r=1/3

3

Write an explicit formula for 5, −10, 20, …

Show hint

The ratio is −2.

Show worked solution
  1. a1=5,r=−2\displaystyle a_1=5,r=-2.
  2. Use an=a1rn−1\displaystyle a_n=a_1r^{n-1}.

Answer: an=5(−2)n−1\displaystyle a_n=5(-2)^{n-1}

4

Find the sum of the first 5 terms of 2, 6, 18, …

Show hint

Use the finite geometric sum.

Show worked solution
  1. a1=2,r=3,n=5\displaystyle a_1=2,r=3,n=5.
  2. S5=2(1−35)/(1−3)\displaystyle S_5=2(1-3^5)/(1-3).
  3. S5=242\displaystyle S_5=242.

Answer: 242\displaystyle 242

5

Find the infinite sum 8+4+2+⋯\displaystyle 8+4+2+\cdots.

Show hint

An infinite geometric sum needs |r| < 1.

Show worked solution
  1. a1=8,r=1/2\displaystyle a_1=8,r=1/2.
  2. S=a1/(1−r)=8/(1/2)\displaystyle S=a_1/(1-r)=8/(1/2).

Answer: 16\displaystyle 16

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A geometric sequence multiplies by the same common ratio r each time. Its nth term is an=a1rn−1\displaystyle a_n=a_1r^{n-1}, and a finite sum is Sn=a1(1−rn)/(1−r)\displaystyle S_n=a_1(1-r^n)/(1-r) for r ≠ 1.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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