Calculate the common difference of an arithmetic sequence from a list of terms or two indexed terms, and find the nth term and sum.
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Common Difference Formula
The common difference is the constant amount added to each term of an arithmetic sequence to get the next term.
d = a(k+1) - a(k)
When you only know two indexed terms, use:
d = (a_n - a_m) / (n - m)
To project forward using a known first term and difference:
a_n = a_1 + (n - 1) * d S_n = n * (a_1 + a_n) / 2
- d = common difference
- a_k = the kth term in the sequence
- a_1 = first term
- a_m, a_n = terms at positions m and n
- n = term position or number of terms
- S_n = sum of the first n terms
The formulas only apply if every adjacent pair has the same difference. If the differences vary across the sequence, it is not arithmetic and no single value of d exists. Positions m and n must be positive integers and cannot be equal.
The calculator has three modes that map to the formulas above:
- From sequence: takes a list of terms, computes every adjacent difference, and returns d only if they all match. It also extends the sequence by three terms.
- Two terms: uses the indexed-term formula to solve for d from any two known positions, then back-solves a_1 and gives you the explicit rule.
- Nth term & sum: applies the projection formulas to give a_n and S_n once you supply a_1, d, and n.
Reference Tables
Quick checks for what d tells you about a sequence and how a_n grows.
| Value of d | Behavior | Example |
|---|---|---|
| d > 0 | Increasing | 2, 5, 8, 11 (d = 3) |
| d = 0 | Constant | 7, 7, 7, 7 |
| d < 0 | Decreasing | 20, 16, 12, 8 (d = -4) |
| Differences vary | Not arithmetic | 1, 2, 4, 8 (geometric) |
| a_1 | d | a_10 | S_10 |
|---|---|---|---|
| 1 | 1 | 10 | 55 |
| 3 | 4 | 39 | 210 |
| 5 | -2 | -13 | -40 |
| 10 | 0.5 | 14.5 | 122.5 |
Worked Examples and FAQ
Example 1: From a sequence. Find d for 6, 9, 12, 15.
9 - 6 = 3, 12 - 9 = 3, 15 - 12 = 3. All match, so d = 3.
Example 2: From two terms. Given a_3 = 11 and a_8 = 31, find d.
d = (31 - 11) / (8 - 3) = 20 / 5 = 4. Then a_1 = 11 - (3 - 1)(4) = 3.
Example 3: Project forward. With a_1 = 3 and d = 4, find a_10 and S_10.
a_10 = 3 + 9(4) = 39. S_10 = 10(3 + 39) / 2 = 210.
Can the common difference be a fraction or negative? Yes. Enter values like -2, 1.5, or 3/4. The calculator accepts simple fractions in any field.
What if the differences are not all equal? The sequence is not arithmetic. The calculator will tell you no single common difference exists and list the differences it found so you can spot where the pattern breaks.
Do I need consecutive terms? No. Use the "Two terms" mode with any two positions. The formula d = (a_n - a_m) / (n - m) handles gaps.
How is this different from the common ratio? Common difference applies to arithmetic sequences where you add d each step. Common ratio applies to geometric sequences where you multiply by r each step.
