Geometric Sequences: Recognize Series

Grade 12 · geometry · 89 practice problems · read aloud

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Geometric Sequences: The Power of Patterns

A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). They are essential for modeling exponential growth and decay, such as in population studies, finance (compound interest), and computer science.

Step-by-Step Guide

  1. Identify the First Term (a₁): This is the starting number of your sequence.
  2. Find the Common Ratio (r): Divide any term by the term immediately before it: r = aₙ / aₙ₋₁.
  3. Use the Explicit Formula: To find the nth term, use: aₙ = a₁ * r⁽ⁿ⁻¹⁾.
  4. Use the Sum Formula: To find the sum of the first n terms, use: Sₙ = a₁(1 - rⁿ) / (1 - r) for r ≠ 1.

Worked Examples

Example 1: Finding a Specific Term

Find the 8th term of the sequence: 3, 6, 12, 24, ...

  1. a₁ = 3
  2. r = 6 / 3 = 2
  3. a₈ = 3 * 2⁽⁸⁻¹⁾ = 3 * 2⁷ = 3 * 128 = 384

Example 2: Finding the Sum

Find the sum of the first 5 terms of: 1000, 500, 250, ...

  1. a₁ = 1000
  2. r = 500 / 1000 = 0.5
  3. S₅ = 1000(1 - 0.5⁵) / (1 - 0.5) = 1000(1 - 0.03125) / 0.5 = 1000(0.96875) / 0.5 = 1937.5

Common Mistakes to Avoid ⚠️

  • Incorrect Exponent: In the explicit formula aₙ = a₁ * r⁽ⁿ⁻¹⁾, the exponent is (n-1), not n. You multiply by the common ratio one fewer time than the term number.
  • Misidentifying 'r': Ensure you divide a term by its preceding term (a₂/a₁), not the other way around. A negative ratio is possible!
  • Sum Formula Error: The sum formula Sₙ = a₁(1 - rⁿ)/(1 - r) is only valid when r ≠ 1. If r = 1, the sequence is constant, and Sₙ = n * a₁.

Tips & Tricks

  • 🧠 Memory Aid: For the term formula, remember "First in Line, Ratio Behind, Power is One Less."
  • 📈 Quick Check: In a growing sequence, r > 1. In a decaying sequence, 0 < r < 1. If the terms alternate signs, r is negative.
  • ✂️ Shortcut: If you know two non-consecutive terms, you can find the ratio by solving for the root. If a₁=2 and a₄=54, then 2 * r³ = 54, so r³=27, r=3.

How to Practice

To master geometric sequences, try these practice strategies:

  • Start by generating sequences from a given a₁ and r.
  • Practice finding the common ratio from both increasing and decreasing sequences.
  • Solve real-world problems involving compound interest or half-life decay.
  • Challenge yourself with problems that require using both the nth-term and the sum formulas together.

Practice problems

6 of the 89, worked through step by step — try them before opening the answer.

1 Is 8, 24, 72, 216... geometric? Find common ratio

Hint: Check if each term can be obtained by multiplying the previous term by the same number. For example, in the sequence 3, 9, 27, 81..., the common ratio is 3.

Show the answer

Answer: 3

  1. Check the ratio between consecutive terms 24 ÷ 8 = 3 72 ÷ 24 = 3 216 ÷ 72 = 3
  2. Since all ratios equal 3, the sequence is geometric
  3. The common ratio r = 3

The answer is 3.

2 Is 4, 12, 36, 108... geometric? Find common ratio

Hint: Check if each term is obtained by multiplying the previous term by the same constant value.

Show the answer

Answer: 3

  1. Check the ratio between consecutive terms 12 ÷ 4 = 3 36 ÷ 12 = 3 108 ÷ 36 = 3
  2. Since all ratios equal 3, this is a geometric sequence
  3. The common ratio r = 3

The answer is 3.

3 Is 3, 15, 75, 375... geometric? Find common ratio

Hint: Check if the ratio between consecutive terms is constant. For example, in a sequence like 2, 10, 50, 250..., the ratio would be 5.

Show the answer

Answer: 5

  1. Check if the sequence is geometric by finding the ratio between consecutive terms.
  2. Calculate 15 ÷ 3 = 5
  3. Calculate 75 ÷ 15 = 5
  4. Calculate 375 ÷ 75 = 5
  5. Since all ratios equal 5, the sequence is geometric with common ratio r = 5.

4 Is 8, 40, 200, 1000... geometric? Find common ratio

Hint: Check if each term divided by the previous term gives the same value

Show the answer

Answer: 5

  1. Check the ratio between consecutive terms 40 ÷ 8 = 5 200 ÷ 40 = 5 1000 ÷ 200 = 5
  2. Since all ratios equal 5, this is a geometric sequence
  3. The common ratio is 5

The answer is 5.

5 Is 12, 36, 108, 324... geometric? Find common ratio

Hint: Check if each term divided by the previous term gives the same value

Show the answer

Answer: 3

  1. Check if the sequence is geometric by finding the ratio between consecutive terms
  2. 36 ÷ 12 = 3
  3. 108 ÷ 36 = 3
  4. 324 ÷ 108 = 3
  5. All ratios equal 3, so the sequence is geometric with common ratio 3 The common ratio is 3.

6 Is 15, 45, 135, 405... geometric? Find common ratio

Hint: Check if each term can be obtained by multiplying the previous term by the same value. For example, in the sequence 4, 12, 36, 108, each term is multiplied by 3 to get the next term.

Show the answer

Answer: 3

  1. Check if the sequence is geometric by finding the ratio between consecutive terms.
  2. 45 ÷ 15 = 3
  3. 135 ÷ 45 = 3
  4. 405 ÷ 135 = 3
  5. Since all ratios equal 3, the sequence is geometric with common ratio r = 3. The common ratio is 3.
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