Recursive Formula for Arithmetic Sequence- How to Write
What Is a Recursive Formula?
A recursive formula defines each term in a sequence using the previous term. Instead of giving you a direct formula to plug in a position number, it tells you: "here's how to get from one term to the next."
For arithmetic sequences specifically, the recursive formula relies on one simple fact: you add the same number (the common difference) each time.
That's it. That's the whole concept. Let's break it down.
Recursive vs Explicit Formulas
Most students learn explicit formulas first. An explicit formula lets you find any term directly using its position.
For the sequence 3, 7, 11, 15, 19... the explicit formula is:
an = 4n - 1
Plug in n = 5, you get a5 = 19. Done.
A recursive formula works differently. It says:
a1 = 3
an = an-1 + 4
To find a5, you need a4. To find a4, you need a3. You work backwards until you hit a1, then build forward.
Recursive formulas are slower for finding single terms. They're useful when you need to understand how a sequence builds step by step.
How to Write a Recursive Formula for an Arithmetic Sequence
Every recursive formula for an arithmetic sequence has exactly two parts:
The Two Parts You Always Need
1. The starting point (base case)
This tells you the value of the first term. Written as a1 = [value].
2. The recurrence relation
This tells you how to get from any term to the next. Written as an = an-1 + d, where d is the common difference.
That's the entire structure. Memorize it.
Step-by-Step: Writing the Formula
Given the sequence: 10, 15, 20, 25, 30...
Step 1: Identify the first term.
a1 = 10
Step 2: Find the common difference.
15 - 10 = 5. The difference is 5.
Step 3: Write the formula.
a1 = 10
an = an-1 + 5
Done. That's your recursive formula.
Another Example
Sequence: -4, 1, 6, 11, 16...
First term: -4
Common difference: 1 - (-4) = 5
Formula:
a1 = -4
an = an-1 + 5
One more.
Sequence: 100, 90, 80, 70...
First term: 100
Common difference: 90 - 100 = -10
Formula:
a1 = 100
an = an-1 - 10
The common difference can be negative. The formula handles it the same way.
Quick Reference Table
| Sequence | First Term (a1) | Common Diff (d) | Recursive Formula |
|---|---|---|---|
| 2, 5, 8, 11... | 2 | 3 | a1 = 2 an = an-1 + 3 |
| 7, 3, -1, -5... | 7 | -4 | a1 = 7 an = an-1 - 4 |
| 0.5, 1.5, 2.5, 3.5... | 0.5 | 1 | a1 = 0.5 an = an-1 + 1 |
Getting Started: Practice Problems
Try writing recursive formulas for these sequences. Answers below.
1. 8, 13, 18, 23, 28...
2. 50, 45, 40, 35, 30...
3. -2, -7, -12, -17, -22...
4. 0.25, 0.5, 0.75, 1.0, 1.25...
Answers:
1. a1 = 8, an = an-1 + 5
2. a1 = 50, an = an-1 - 5
3. a1 = -2, an = an-1 - 5
4. a1 = 0.25, an = an-1 + 0.25
Common Mistakes to Avoid
- Forgetting the base case. The formula is incomplete without a1. Without it, you have no starting point.
- Getting the sign wrong on the common difference. Always subtract: larger term minus smaller term. If the sequence decreases, your difference is negative.
- Writing an = an + d. This is circular and wrong. The left side should be the current term, the right side should reference the previous term.
- Confusing the subscripts. an-1 means "the term right before the one you're finding." an means "the term at position n."
When Recursive Formulas Actually Matter
You won't use recursive formulas to find the 50th term of a sequence. Use the explicit formula for that. Recursive formulas matter in:
- Computer programming and algorithms
- Calculus (defining sequences and series)
- Financial calculations (compound interest builds recursively)
- Physics problems involving step-by-step changes
The pattern is simple: find a1, find d, plug into the two-part structure. Practice with five sequences and you'll have it locked down.