Find an Explicit Formula For- Sequence and Series Tutorial
What Is an Explicit Formula for a Sequence?
An explicit formula gives you the nth term of a sequence directly. No adding previous terms. No guesswork. You plug in n, you get the answer.
That's it. That's the whole point.
Compare this to a recursive formula, which defines each term using the previous one. Recursive is useful for definitions. Explicit is useful for calculations.
If you're given a sequence like 3, 7, 11, 15, ... and asked to find the 50th term, you don't want to list all 50 terms. You want a formula.
The Two Formulas You Must Know First
Arithmetic Sequences
An arithmetic sequence adds the same number each step. The explicit formula is:
an = a1 + (n - 1)d
Where:
- a1 is the first term
- d is the common difference
- n is the term position
Example: 5, 8, 11, 14, ...
a1 = 5, d = 3
an = 5 + (n - 1)(3) = 5 + 3n - 3 = 3n + 2
Geometric Sequences
A geometric sequence multiplies by the same number each step. The explicit formula is:
an = a1 · r(n-1)
Where:
- a1 is the first term
- r is the common ratio
- n is the term position
Example: 2, 6, 18, 54, ...
a1 = 2, r = 3
an = 2 · 3(n-1)
How to Find an Explicit Formula: Step by Step
When you're given a sequence and need to find the explicit formula, follow this process:
Step 1: Check if the differences are constant
Calculate the difference between consecutive terms. If the difference stays the same, it's arithmetic. That's your formula.
Sequence: 10, 17, 24, 31
17 - 10 = 7
24 - 17 = 7
31 - 24 = 7 ✓
Formula: an = 10 + (n - 1)(7) = 7n + 3
Step 2: Check if the ratios are constant
Calculate the ratio between consecutive terms. If the ratio stays the same, it's geometric.
Sequence: 3, 15, 75, 375
15 ÷ 3 = 5
75 ÷ 15 = 5
375 ÷ 75 = 5 ✓
Formula: an = 3 · 5(n-1)
Step 3: When it's neither
Some sequences aren't arithmetic or geometric. You need to look for other patterns.
Sequence: 1, 4, 9, 16, 25
These are perfect squares: 12, 22, 32, 42, 52
Formula: an = n2
Sequence: 1, 1, 2, 3, 5, 8
This is the Fibonacci sequence. No simple explicit formula exists. You'd need to use Binet's formula with the golden ratio, which is messy and rarely what instructors expect.
Sequences vs. Series: Don't Mix These Up
A sequence is a list of numbers: 2, 4, 6, 8
A series is the sum of those numbers: 2 + 4 + 6 + 8
When you see "find the explicit formula for the series," they usually mean find the explicit formula for the sequence of partial sums, or find a formula for the nth term of the underlying sequence.
For arithmetic series, the sum formula is:
Sn = n(a1 + an) / 2
For geometric series, the sum formula is:
Sn = a1(1 - rn) / (1 - r) for r ≠ 1
Common Patterns to Recognize
- Linear patterns: differences are constant → arithmetic
- Exponential patterns: ratios are constant → geometric
- Quadratic patterns: second differences are constant → formula involves n2
- Polynomial patterns: higher-order differences constant → match degree to difference level
- Factorial patterns: n! appears → look for factorial notation
- Alternating signs: multiply by (-1)n or (-1)n+1
Quick Reference Table
| Sequence Type | Pattern | Explicit Formula |
|---|---|---|
| Arithmetic | Constant difference | an = a1 + (n-1)d |
| Geometric | Constant ratio | an = a1 · rn-1 |
| Quadratic | Second differences constant | an = an2 + bn + c |
| Fibonacci-style | Sum of previous terms | No simple explicit formula (use Binet's formula) |
Getting Started: Practice Problems
Find the explicit formula for each sequence:
1. 7, 12, 17, 22, 27, ...
Difference is 5. a1 = 7.
Answer: an = 7 + (n-1)(5) = 5n + 2
2. 100, 50, 25, 12.5, ...
Ratio is 0.5. a1 = 100.
Answer: an = 100 · (0.5)n-1
3. 3, 6, 11, 18, 27, ...
First differences: 3, 5, 7, 9 (not constant)
Second differences: 2, 2, 2 (constant!) → quadratic
Answer: an = n2 + 2
What to Do When You're Stuck
Look at the first few terms. Try to spot the pattern with simple operations before assuming complexity.
Ask yourself:
- Am I just adding the same number?
- Am I just multiplying by the same number?
- Is this a known pattern (squares, cubes, factorials)?
- Do the differences follow a pattern?
If none of that works, check if the sequence involves alternating signs or a combination of operations.
Most textbook problems use arithmetic or geometric sequences. If your sequence doesn't fit either, look for quadratic or factorial patterns. Those cover 90% of what you'll encounter.