11.3Geometric Sequences
Apply the explicit formula aₙ = a₁ · rⁿ⁻¹ and recursive formula to model exponential growth, compound interest, and bouncing ball problems.
Geometric sequences describe exponential growth and decay — compound interest, population models, radioactive decay, and fractal geometry all rely on geometric patterns. They connect directly to exponential functions and calculus.
Essential Question
How can we model situations where quantities multiply by a constant factor — and predict any term without listing them all?
Overview
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio r.
- Common ratio: r = aₙ / aₙ₋₁ (must be constant)
- Explicit formula: aₙ = a₁ · rn−1
- Recursive formula: aₙ = r · aₙ₋₁
- Finding r from two terms: r = (aₙ / a₁)1/(n−1)
- Geometric mean between a and b: √(ab)
Geometric sequences model exponential growth and decay: compound interest, population growth, radioactive decay, and bouncing ball heights.
Worked Examples
Find the first 5 terms of the geometric sequence with a₁ = 3 and r = 2.
a₁ = 3
a₂ = 3 · 2 = 6
a₃ = 6 · 2 = 12
a₄ = 12 · 2 = 24
a₅ = 24 · 2 = 48
Find a₈ for the geometric sequence with a₁ = 5 and r = −2.
Use the explicit formula: aₙ = a₁ · rⁿ⁻¹
a₈ = 5 · (−2)⁸⁻¹ = 5 · (−2)⁷
(−2)⁷ = −128
a₈ = 5 · (−128) = −640
Find the common ratio r and the explicit formula given a₂ = 6 and a₅ = 162.
From a₂ to a₅ there are 5 − 2 = 3 steps, so a₅ = a₂ · r³
162 = 6 · r³ → r³ = 162 / 6 = 27
r = ∛27 = 3
Find a₁: a₂ = a₁ · r → 6 = a₁ · 3 → a₁ = 2
Explicit formula: aₙ = 2 · 3ⁿ⁻¹
Insert 2 geometric means between 4 and 108.
We need the sequence: 4, g₁, g₂, 108 — four terms total.
a₁ = 4, a₄ = 108, so a₄ = a₁ · r³
108 = 4 · r³ → r³ = 27 → r = 3
g₁ = 4 · 3 = 12
g₂ = 12 · 3 = 36
$1,000 is invested at 6% annual compound interest. Find the value after 10 years.
Use A = P(1 + r)ⁿ where P = 1000, r = 0.06, n = 10
A = 1000 · (1.06)¹⁰
(1.06)¹⁰ ≈ 1.7908
A ≈ 1000 · 1.7908 = 1790.85
Guided Practice
Find the first 5 terms of the geometric sequence with a₁ = 4 and r = 3.
Hint: Multiply each term by 3 to get the next. Start with 4.
Find a₆ for the geometric sequence with a₁ = 2 and r = −3.
Hint: Use aₙ = a₁ · rⁿ⁻¹. Substitute n = 6 and be careful with the negative sign: (−3)⁵.
Find r and the explicit formula given a₁ = 5 and a₄ = 40.
Hint: Use a₄ = a₁ · r³. Solve for r³ first, then take the cube root.
Insert 1 geometric mean between 3 and 75.
Hint: You need 3, g, 75. That's 3 terms, so a₃ = a₁ · r². Solve for r, then find g = a₁ · r.
A town has a population of 10,000 and grows 5% per year. Find the population after 6 years.
Hint: This is geometric with a₁ = 10,000 and r = 1.05. Use aₙ = a₁ · rⁿ⁻¹ with n = 7 (after 6 years means the 7th term).
Common Mistakes
Common Mistakes
Using aₙ = a₁ · rⁿ instead of aₙ = a₁ · rⁿ⁻¹ (off-by-one error).
The exponent is n − 1 because the first term has r⁰ = 1 (no multiplication yet).
Forgetting the negative sign when r is negative: (−2)⁷ = 128.
Odd powers of a negative number are negative: (−2)⁷ = −128.
Confusing arithmetic (add d) with geometric (multiply r) sequences.
Check: if ratios are constant → geometric. If differences are constant → arithmetic.
When inserting k geometric means, using r² instead of rᵏ⁺¹.
Inserting k means creates k + 2 total terms. Use aₖ₊₂ = a₁ · rᵏ⁺¹ to find r.
Math Tips
Math Tips
To check if a sequence is geometric, divide consecutive terms — the ratio must be the same every time.
If r > 1, the sequence grows. If 0 < r < 1, it decays. If r < 0, terms alternate in sign.
The geometric mean of two numbers a and b is √(ab) — it is the single term that fits between them geometrically.
Compound interest A = P(1 + r)ⁿ is just the explicit formula aₙ = a₁ · rⁿ with a₁ = P and ratio = (1 + r).
When r = 1, every term equals a₁ — a constant sequence (technically geometric but trivial).
Key Vocabulary
Geometric Sequence
A sequence where each term is found by multiplying the previous term by a constant ratio r.
Common Ratio (r)
The constant multiplier between consecutive terms: r = aₙ / aₙ₋₁.
Explicit Formula
A formula that gives any term directly: aₙ = a₁ · rⁿ⁻¹.
Recursive Formula
A formula that defines each term using the previous one: aₙ = r · aₙ₋₁.
Geometric Mean
The value g such that a, g, b is geometric; g = √(ab).
Compound Interest
Interest calculated on both principal and accumulated interest: A = P(1 + r)ⁿ.
Interactive Practice — 1 Questions
Which formula correctly gives the nth term of a geometric sequence?