Recursive Formula for Geometric Sequences Explained
What Is a Geometric Sequence?
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. That's it. No addition, no fancy patterns—just multiplication.
Examples:
- 2, 6, 18, 54, 162... (multiply by 3 each time)
- 100, 50, 25, 12.5... (multiply by 0.5 each time)
- 3, -9, 27, -81... (multiply by -3 each time)
The common ratio can be positive, negative, a fraction, or greater than 1. It doesn't matter what it is—what matters is that it never changes.
What Is a Recursive Formula?
A recursive formula defines each term using the term before it. Instead of giving you a formula to plug in n and get the answer directly, it says: "to find the next term, take what you just found and multiply by the ratio."
Every recursive formula needs two things:
- The first term (called a₁ or the initial condition)
- A rule that tells you how to get from one term to the next
Without the first term, you have nothing to start with. Without the rule, you have no way to continue. Both are non-negotiable.
Writing Recursive Formulas for Geometric Sequences
The template is embarrassingly simple once you see it:
a₁ = first term
aₙ = aₙ₋₁ × r (where r is the common ratio)
That's the whole thing. Let me show you how this works with real numbers.
Example 1: Positive Common Ratio
Sequence: 5, 15, 45, 135...
What's the ratio? 15 ÷ 5 = 3. The ratio is 3.
Recursive formula:
a₁ = 5
aₙ = aₙ₋₁ × 3
To find a₄: a₃ × 3 = 45 × 3 = 135. Done.
Example 2: Fraction as Common Ratio
Sequence: 64, 32, 16, 8...
What's the ratio? 32 ÷ 64 = 0.5 (or 1/2).
Recursive formula:
a₁ = 64
aₙ = aₙ₋₁ × 0.5
Or if you prefer fractions: aₙ = aₙ₋₁ × ½
Example 3: Negative Common Ratio
Sequence: 2, -6, 18, -54...
What's the ratio? -6 ÷ 2 = -3.
Recursive formula:
a₁ = 2
aₙ = aₙ₋₁ × (-3)
Signs flip every term. That's fine. Just keep multiplying by -3.
How To: Find Specific Terms Step by Step
Let's say you need to find the 7th term of a sequence with a₁ = 4 and r = 3.
Step 1: Write down what you know. a₁ = 4, r = 3.
Step 2: Build the sequence term by term.
- a₁ = 4
- a₂ = 4 × 3 = 12
- a₃ = 12 × 3 = 36
- a₄ = 36 × 3 = 108
- a₅ = 108 × 3 = 324
- a₆ = 324 × 3 = 972
- a₇ = 972 × 3 = 2,916
The 7th term is 2,916.
This is the main weakness of recursive formulas: to find a₇, you need a₆. To find a₆, you need a₅. You can't skip steps. If you need the 50th term, you're doing this 49 times. That's not efficient—and that's exactly why explicit formulas exist.
Recursive vs. Explicit Formulas
Here's the comparison nobody asked for but everyone needs:
| Feature | Recursive | Explicit |
|---|---|---|
| Uses previous terms? | Yes | No |
| Can skip to any term? | No—you must calculate all terms before it | Yes—just plug in n |
| Formula structure | a₁ = first term; aₙ = aₙ₋₁ × r | aₙ = a₁ × rⁿ⁻¹ |
| Best for | Sequential problems, programming, seeing patterns | Finding specific terms quickly |
| Effort to find a₁₀₀ | Calculate 99 multiplications | One calculation |
Neither is better. They do different jobs. Recursive is how computers think. Explicit is how humans save time.
Converting Between the Two
If you have a recursive formula and need the explicit version (or vice versa), here's how:
Recursive to Explicit
Given: a₁ = 5, aₙ = aₙ₋₁ × 3
Find a₂: 5 × 3 = 15
Find a₃: 15 × 3 = 45
Find a₄: 45 × 3 = 135
Pattern: each term is the first term multiplied by 3 raised to (n-1).
Explicit formula: aₙ = 5 × 3ⁿ⁻¹
Explicit to Recursive
Given: aₙ = 2 × 4ⁿ⁻¹
First term: a₁ = 2 × 4⁰ = 2 × 1 = 2
Ratio: 4 (the base of the exponent)
Recursive formula: a₁ = 2, aₙ = aₙ₋₁ × 4
Common Mistakes to Avoid
- Forgetting the first term. You cannot write a recursive formula without stating a₁. Period.
- Using the wrong ratio. Always check: divide any term by the previous term. If you get the same answer every time, that's your ratio.
- Writing aₙ = aₙ₋₁ instead of aₙ = aₙ₋₁ × r. The multiplication is what makes it geometric. Without it, you're describing an arithmetic sequence.
- Confusing n and aₙ. n is the position (1st, 2nd, 3rd...). aₙ is the value at that position.
- Rounding the ratio. If the ratio is 1/3, keep it as 1/3. Don't round to 0.33—your answers will drift.
Quick Reference Template
When you see a geometric sequence and need to write its recursive formula:
- Identify the first term → this is a₁
- Divide any term by the previous term → this is r
- Write: a₁ = [first term]
- Write: aₙ = aₙ₋₁ × [ratio]
That's the entire process. There's no hidden trick. Identify two numbers, write them down in the right spots, and you're done.