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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(3^4).
  5. The term is 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(3^4).
  5. The term is 243.
  6. Check the result by substituting or estimating: the response should match 243 and make sense in the original problem.

Answer: 243

2. Find a_6 for the geometric sequence with a_1 = 4 and r = 4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find a_6 for the geometric sequence with a_1 = 4 and r = 4.
  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_6 = 4(4^5).
  5. The term is 4096.
  6. Check the result by substituting or estimating: the response should match 4096 and make sense in the original problem.

Answer: 4096

3. Find a_4 for the geometric sequence with a_1 = 5 and r = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_4 for 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_n = a_1 r^(n - 1).
  4. a_4 = 5(2^3).
  5. The term is 40.
  6. Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.

Answer: 40

4. Find a_5 for the geometric sequence with a_1 = 2 and r = 3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_5 for the geometric sequence with a_1 = 2 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 = 2(3^4).
  5. The term is 162.
  6. Check the result by substituting or estimating: the response should match 162 and make sense in the original problem.

Answer: 162

5. Find a_6 for the geometric sequence with a_1 = 3 and r = 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find a_6 for the geometric sequence with a_1 = 3 and r = 4.
  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_6 = 3(4^5).
  5. The term is 3072.
  6. Check the result by substituting or estimating: the response should match 3072 and make sense in the original problem.

Answer: 3072

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