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Geometric Sequences

A free Precalculus lesson from the “Sequences, Series, and Discrete Models” unit, with a worked example and practice problems including step-by-step solutions.

Geometric sequences multiply by the same factor each step. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Sequences and series connect repeated patterns, finance, computer loops, and discrete approximations.

Worked example

Problem. Find a_5 for the geometric sequence with a_1 = 3 and r = 3.

  1. Worked Example: First identify exactly what the question is asking: Find a_5 for the geometric sequence with a_1 = 3 and r = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 * r^(n - 1).
  4. a_5 = 3 * 3^(5 - 1) = 3 * 3^4.
  5. So a_5 = 243.
  6. Check the result by substituting or estimating: the response should match 243 and make sense in the original problem.

Answer: 243

Practice problems

1. Find a_5 for the geometric sequence with a_1 = 3 and r = 3.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find a_5 for the geometric sequence with a_1 = 3 and r = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_n = a_1 * r^(n - 1).
  4. a_5 = 3 * 3^(5 - 1) = 3 * 3^4.
  5. So a_5 = 243.
  6. Check the result by substituting or estimating: the response should match 243 and make sense in the original problem.

Answer: 243

2. A geometric sequence has a_1 = 4 and r = 4. Find the second term a_2.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A geometric sequence has a_1 = 4 and r = 4. Find the second term a_2.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_2 = a_1 * r.
  4. a_2 = 4 * 4.
  5. So a_2 = 16.
  6. Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.

Answer: 16

3. Find the third term a_3 of the geometric sequence with a_1 = 5 and r = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the third term a_3 of the geometric sequence with a_1 = 5 and r = 2.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use a_3 = a_1 * r^2.
  4. a_3 = 5 * 2^2 = 5 * 4.
  5. So a_3 = 20.
  6. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

4. Find the sum of the first 3 terms of the geometric sequence with a_1 = 2 and r = 3.

Show solution
  1. This is a finite geometric series; use S_3 = a_1 * (r^3 - 1) / (r - 1).
  2. S_3 = 2 * (3^3 - 1) / (3 - 1) = 2 * 26 / 2 = 2 + 6 + 18.
  3. So the sum is 26.

Answer: 26

5. Two consecutive terms of a geometric sequence are 768 and 3072. Find the common ratio r.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Two consecutive terms of a geometric sequence are 768 and 3072. Find the common ratio r.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide a term by the term before it: r = a_(n+1) / a_n.
  4. r = 3072 / 768.
  5. So r = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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