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H42 • Lesson 64 of 105

Sequences - Geometric

Geometric sequences, common ratio, formulas

High School Essentials • 9-12

Prerequisites: H41

Key Concepts

  • geometric
  • sequence
  • ratio

Geometric Sequences

While arithmetic sequences grow by adding, geometric sequences grow by multiplying. Each term is obtained by multiplying the previous term by a fixed value called the common ratio. This multiplicative pattern appears in compound interest, population dynamics, fractal geometry, and signal processing.

Identifying Geometric Sequences

A sequence is geometric if the ratio between consecutive terms is always the same:

r = an+1 / an  (constant for all n)

Geometric

3, 6, 12, 24, ... (r = 2)

100, 50, 25, 12.5, ... (r = 0.5)

NOT Geometric

2, 5, 8, 11, ... (constant difference, not ratio)

1, 4, 9, 16, ... (perfect squares)

Formulas

FormulaExpressionUse
Explicit (nth term)an = a1 · r(n-1)Jump to any term
Recursivean = an-1 · r, with a1 givenEach term from the previous
Partial sum (r ≠ 1)Sn = a1(1 - rn) / (1 - r)Sum of the first n terms
Infinite sum (|r| < 1)S = a1 / (1 - r)Sum of all terms when series converges

Worked Example 1 -- Finding Terms

Find the 8th term of the sequence 5, 15, 45, 135, ...

  • Identify: a1 = 5, r = 15/5 = 3.
  • Apply: a8 = 5 · 37 = 5 · 2187 = 10,935.

Worked Example 2 -- Partial Sum

Find the sum of the first 6 terms of: 4, 12, 36, 108, ...

  • a1 = 4, r = 3, n = 6.
  • S6 = 4(1 - 36) / (1 - 3) = 4(1 - 729) / (-2) = 4(-728)/(-2) = 1,456.

Worked Example 3 -- Infinite Series

Find the sum: 18 + 6 + 2 + 2/3 + ...

  • a1 = 18, r = 6/18 = 1/3. Since |1/3| < 1, the series converges.
  • S = 18 / (1 - 1/3) = 18 / (2/3) = 18 × 3/2 = 27.

Repeating Decimals as Infinite Series

The repeating decimal 0.333... = 3/10 + 3/100 + 3/1000 + ... is an infinite geometric series with a1 = 0.3 and r = 0.1. Sum = 0.3/(1 - 0.1) = 0.3/0.9 = 1/3. This technique proves that every repeating decimal is a fraction.

Common Mistake

The infinite sum formula S = a1/(1 - r) only works when |r| < 1. If |r| ≥ 1, the terms do not shrink toward zero and the sum diverges -- it grows without bound.

Practice Problems

  1. Find the 6th term of: 2, -6, 18, -54, ...
    Show Solution

    a1 = 2, r = -3. a6 = 2 · (-3)5 = 2 · (-243) = -486.

  2. Write the explicit formula for: 1000, 500, 250, 125, ...
    Show Solution

    a1 = 1000, r = 0.5. an = 1000 · (0.5)n-1.

  3. Find the sum of the first 5 terms of: 1, 1/2, 1/4, 1/8, 1/16.
    Show Solution

    S5 = 1(1 - (1/2)5) / (1 - 1/2) = (1 - 1/32)/(1/2) = (31/32)/(1/2) = 31/16 = 1.9375.

  4. Find the sum of the infinite series: 24 + 12 + 6 + 3 + ...
    Show Solution

    a1 = 24, r = 1/2. S = 24/(1 - 1/2) = 24/(1/2) = 48.

  5. A ball is dropped from 80 feet and bounces to 60% of its previous height each time. What is the total distance it travels (up and down) before coming to rest?
    Show Solution

    Down distances: 80, 48, 28.8, ... (a1 = 80, r = 0.6). Up distances: 48, 28.8, ... (a1 = 48, r = 0.6). Total = 80/(1 - 0.6) + 48/(1 - 0.6) = 200 + 120 = 320 feet. Alternatively: first drop of 80, then twice each bounce: total = 80 + 2(48)/(1 - 0.6) = 80 + 240 = 320 feet.

Summary

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