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Mom's Basement University

Geometric Sequences

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what a geometric sequence is, how to find the common ratio, and how to write both the explicit and recursive formulas.

The Concept

Section titled “The Concept”

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio (r).

The diagram shows the sequence 3, 6, 12, 24, 48. Each arc shows the common ratio r = ×2 being multiplied to get the next term. Every geometric sequence works this way. You start with a first term and keep multiplying by the same number.

Examples:

  • 3, 6, 12, 24, 48, … → common ratio r = 2
  • 80, 40, 20, 10, … → common ratio r = 1/2

Explicit formula (gives the nth term directly):

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

Recursive formula (gives the next term from the previous one):

A recursive formula defines each term using the term before it. Instead of jumping straight to any term like the explicit formula does, you build the sequence one step at a time. To find the 5th term you need the 4th, to find the 4th you need the 3rd, and so on. That’s why a recursive formula always needs a starting value (a₁) to get going.

an=an−1⋅r(n≥2)a_n = a_{n-1} \cdot r \quad (n \ge 2)

This says: “to get any term, take the previous term and multiply by r.”

Growth vs. decay:

  • If |r| > 1, the terms grow larger (growth)
  • If 0 < |r| < 1, the terms get smaller (decay)

Worked Example

Section titled “Worked Example”

1. Find the common ratio and the 7th term of 5, 15, 45, 135, …

Common ratio: r = 15 ÷ 5 = 3.

a7=5⋅36=5⋅729=3,645a_7 = 5 \cdot 3^{6} = 5 \cdot 729 = 3{,}645

2. Write the explicit formula for 64, 32, 16, 8, …

a₁ = 64, r = 1/2.

an=64⋅(12)n−1a_n = 64 \cdot \left(\frac{1}{2}\right)^{n-1}

3. Given the recursive formula aₙ = aₙ₋₁ · (−2) with a₁ = 3, find a₄

a2=3×(−2)=−6a3=−6×(−2)=12a4=12×(−2)=−24\begin{aligned} a_2 &= 3 \times (-2) = -6 \\[0.5em] a_3 &= -6 \times (-2) = 12 \\[0.5em] a_4 &= 12 \times (-2) = -24 \end{aligned}

Real-World Application

Section titled “Real-World Application”

Geometric sequences model many real-life situations:

  • Compound interest and investment growth
  • Population growth or decay (when the rate is a constant percentage)
  • Depreciation of assets by a fixed percentage each year
  • Half-life of radioactive substances
  • Successive percentage discounts or increases

Example: an investment of 1000 dollars grows at 6% per year compounded annually. The value after n years follows a geometric sequence with r = 1.06.

Quiz

Section titled “Quiz”
What is the common ratio in 4, 12, 36, 108, …?
The explicit formula for a geometric sequence is:
Find the 6th term: $a_1 = 5$, $r = 2$
A geometric sequence has a constant:
A geometric sequence has $a_1 = 3$ and $r = \frac{1}{3}$. What is $a_4$?