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Geometric Sequences

A free Algebra I lesson from the “Exponential Functions and Sequences” unit, with a worked example and practice problems including step-by-step solutions.

A geometric sequence multiplies by the same number r (the common ratio) to get from one term to the next. The explicit formula for the n-th term is a_n = a_1 * r^(n - 1). If r > 1 the sequence grows; if 0 < r < 1 it shrinks; if r is negative the signs alternate.

What you'll learn

Why it matters: Compound interest, population doubling, radioactive halving, and computer memory sizing (each doubling generation) are geometric.

Worked example

Problem. For the sequence 2, 6, 18, 54, ..., find r and a_5.

  1. Ratio: 6 / 2 = 3.
  2. Apply the formula: a_5 = 2 * 3^4 = 2 * 81 = 162.

Answer: r = 3, a_5 = 162

Practice problems

1. For 1, 2, 4, 8, ..., find r.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 1, 2, 4, 8, ..., find r.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 2 / 1 = 2.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. For 1, 2, 4, 8, ..., find a_6.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 1, 2, 4, 8, ..., find a_6.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. a_6 = 1 * 2^5 = 32.
  4. Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.

Answer: 32

3. For 81, 27, 9, 3, ..., find r as a fraction.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For 81, 27, 9, 3, ..., find r as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. 27 / 81 = 1/3.
  4. Check the result by substituting or estimating: the response should match 1/3 and make sense in the original problem.

Answer: 1/3

4. For 5, 10, 20, 40, ..., find a_5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 5, 10, 20, 40, ..., find a_5.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. r = 2; a_5 = 5 * 2^4 = 80.
  4. Check the result by substituting or estimating: the response should match 80 and make sense in the original problem.

Answer: 80

5. For 100, 50, 25, ..., find r as a fraction.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For 100, 50, 25, ..., find r as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. 50 / 100 = 1/2.
  4. Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.

Answer: 1/2

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