How to Do Arithmetic Sequences- Complete Guide
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers where the difference between consecutive terms stays the same. That constant difference is called the common difference, denoted by d.
Examples:
- 2, 5, 8, 11, 14... (d = 3)
- 100, 90, 80, 70, 60... (d = -10)
- 7, 7, 7, 7, 7... (d = 0)
If you spot a pattern where you keep adding or subtracting the same number, you're looking at an arithmetic sequence.
The Core Formula
The nth term of an arithmetic sequence follows this rule:
an = a1 + (n - 1)d
Where:
- an = the term you want to find
- a1 = the first term
- n = the position number
- d = the common difference
That's it. Memorize this formula or write it down. You'll use it constantly.
How to Find the nth Term
Step 1: Identify the first term (a1)
Look at the sequence. The very first number is your a1.
Step 2: Find the common difference (d)
Subtract any term from the next term. Do this twice to confirm it's consistent.
Example: 4, 9, 14, 19...
9 - 4 = 5
14 - 9 = 5
Confirmed: d = 5
Step 3: Plug into the formula
Find the 20th term of 4, 9, 14, 19...
a20 = 4 + (20 - 1) × 5
a20 = 4 + 95
a20 = 99
How to Find the Sum of an Arithmetic Sequence
When you need the sum of the first n terms, use the sum formula:
Sn = (n/2)(a1 + an)
Or the version that doesn't require finding an first:
Sn = (n/2)[2a1 + (n - 1)d]
Practical Example
Find the sum of the first 50 positive integers: 1, 2, 3, 4... 50
a1 = 1, an = 50, n = 50
S50 = (50/2)(1 + 50)
S50 = 25 × 51
S50 = 1275
Arithmetic vs. Geometric Sequences
People mix these up constantly. Here's the difference:
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Pattern | Add/subtract the same number | Multiply/divide by the same number |
| Key value | Common difference (d) | Common ratio (r) |
| Example | 3, 7, 11, 15... | 3, 6, 12, 24... |
| Formula | an = a1 + (n-1)d | an = a1 × rn-1 |
Arithmetic grows linearly. Geometric grows exponentially. Know which one you're working with.
Common Difference Can Be Negative
A sequence doesn't have to increase. If d is negative, the terms decrease.
Example: 100, 85, 70, 55, 40...
d = 85 - 100 = -15
Find the 15th term:
a15 = 100 + (15 - 1)(-15)
a15 = 100 - 210
a15 = -110
The sequence crosses zero and goes negative. That's fine.
How to Identify an Arithmetic Sequence
Quick test: subtract each term from the next one. If you get the same number every time, it's arithmetic.
Check: 17, 23, 29, 35
- 23 - 17 = 6
- 29 - 23 = 6
- 35 - 29 = 6
Confirmed. This is arithmetic with d = 6.
Check: 2, 4, 8, 16
- 4 - 2 = 2
- 8 - 4 = 4
- 16 - 8 = 8
Not the same. This is not arithmetic—it's geometric.
Getting Started: Quick Reference
When you face an arithmetic sequence problem, follow this checklist:
- Write down what you know — first term, position, common difference
- Find d if it's not given — subtract consecutive terms
- Choose your formula — nth term for a single term, sum formula for totals
- Plug in the numbers — double-check your arithmetic
- Write the answer — include the position number (e.g., "the 12th term is 47")
Why This Matters
Arithmetic sequences show up in finance (loan payments), computer science (array indexing), physics (constant acceleration problems), and standardized testing. The formula is straightforward. The execution is where people mess up—usually by forgetting to subtract 1 from n, or by using the wrong sign for d.
Practice with 10 problems. Check your answers. The pattern becomes automatic after the third or fourth one.