Find the fifth term of 3, 6, 12, … .
- The ratio is 2.
- a₅ = 3 × 2⁴ = 3 × 16.
Answer: 48
Geometric Sequences and Series Worksheets
Find terms and sums with explicit starting indices and convergence conditions.
Learn the skill
Geometric sequences and finite sums
A geometric sequence multiplies by the same ratio at every step. If the first term is a₁ and the common ratio is r, its nth term is a₁r^(n−1). A series is the sum of terms, not an individual term.
A method you can use
- Divide consecutive nonzero terms to check for a constant ratio.
- Use n − 1 multiplications to get from the first term to the nth term.
- For a finite sum with r ≠ 1, use S_n = a₁(1 − r^n)/(1 − r); if r = 1, the sum is n × a₁.
Worked examples
Find the fourth term of 16, 8, 4, … .
- The ratio is 1/2.
- a₄ = 16 × (1/2)³.
Answer: 2
Find the sum of the first four terms of 2, 6, 18, … .
- The first four terms are 2, 6, 18, 54.
- Add 2 + 6 + 18 + 54, or use the finite-sum formula.
Answer: 80
Try it yourself
Solve these before opening the answers. The printable sets below give you more practice.
1. Find the fourth term of 5, 10, 20, … .
Show answer and explanation
40
Multiply the first term by 2³.
2. Find the sum of the first three terms of 3, 9, 27, … .
Show answer and explanation
39
Add 3 + 9 + 27.
Before you begin
Factor polynomials, use exponent laws and solve equations with domain restrictions.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
- Easy: Start with the preview and follow the worked example.
- Medium: Practice the same exercise types with a fresh numbered set.
- Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
A geometric sequence has a₁ = 3 and common ratio 3. Find a6.
2
A geometric sequence has a₁ = 9 and common ratio 2. Find the sum of its first 5 terms.
3
A geometric sequence has a₁ = 10 and common ratio 4. Find a3.
4
A geometric sequence has a₁ = 8 and common ratio 3. Find the sum of its first 5 terms.
5
A geometric sequence has a₁ = 8 and common ratio 5. Find a4.
6
A geometric sequence has a₁ = 12 and common ratio 5. Find the sum of its first 4 terms.
Answer Key
1. 7292. 2793. 1604. 9685. 10006. 1872