Arithmetic Sequence: worked example
| First term (a₁) | 3 |
|---|---|
| Common difference (d) | 4 |
| Number of terms (n) | 10 |
| Last term (aₙ) | 39 |
| Sum of all terms | 210 |
Arithmetic Sequence: formula
Enter the first term, the common difference, which is the amount added each step and can be negative, and the number of terms. The calculator gives the last term using a1 plus n minus 1 times d, and the sum using n times the average of the first and last terms. Whole numbers or decimals work.
Check that the number of terms is a positive whole number.
Arithmetic Sequence: changing First term (a₁)
| First term (a₁) | Last term (aₙ) | Sum of all terms |
|---|---|---|
| 1 | 37 | 190 |
| 2 | 38 | 200 |
| 3 (example) | 39 | 210 |
| 4 | 40 | 220 |
| 5 | 41 | 230 |
Arithmetic Sequence: compare input ranges
| Input | Tested from | To | Last term (aₙ): low input | Last term (aₙ): high input | Swing |
|---|---|---|---|---|---|
| Common difference (d) | 3 | 5 | 30 | 48 | 18 (46%) |
| Number of terms (n) | 9 | 11 | 35 | 43 | 8 (21%) |
| First term (a₁) | 2 | 4 | 38 | 40 | 2 (5.1%) |
Arithmetic Sequence: individual input comparisons
| Input | Value | Last term (aₙ) | Sum of all terms |
|---|---|---|---|
| First term (a₁) | 1 | 37 | 190 |
| 2 | 38 | 200 | |
| 3 | 39 | 210 | |
| 4 | 40 | 220 | |
| 5 | 41 | 230 | |
| Common difference (d) | 2 | 21 | 120 |
| 3 | 30 | 165 | |
| 4 | 39 | 210 | |
| 5 | 48 | 255 | |
| 6 | 57 | 300 | |
| Number of terms (n) | 8 | 31 | 136 |
| 9 | 35 | 171 | |
| 10 | 39 | 210 | |
| 11 | 43 | 253 | |
| 12 | 47 | 300 |
Arithmetic Sequence: combined input changes
| First term (a₁) / Common difference (d) | 2 | 4 | 6 |
|---|---|---|---|
| 1 | 19 | 37 | 55 |
| 2 | 20 | 38 | 56 |
| 3 | 21 | 39 | 57 |
| 4 | 22 | 40 | 58 |
| 5 | 23 | 41 | 59 |
Arithmetic Sequence: which assumptions move the result?
Arithmetic Sequence: values and definitions
| Quantity and definition | Value used |
|---|---|
| First term (a₁) — The starting value of the sequence. | 3 |
| Common difference (d) — The fixed amount added at every step. Negative for a falling sequence. | 4 |
| Number of terms (n) — How many terms to include, from 1 to 1,000. | 10 |
| Last term (aₙ) | 39 |
| Sum of all terms | 210 |
Arithmetic Sequence: half and double Common difference (d)
| Common difference (d) | Last term (aₙ) |
|---|---|
| 2 | 21 |
| 4 | 39 |
| 8 | 75 |
Common error: Counting the steps instead of the terms. Ten terms have nine steps between them, so the last term is the first plus nine differences, not ten.
The pattern must have a constant difference to be arithmetic. Growth by a constant percentage is geometric, not arithmetic. A negative difference produces a decreasing sequence.
The formulas assume a finite number of terms, and infinite arithmetic sums do not converge unless the terms are zero.
A negative starting term does not change the rule
The assumed list −2, 1, 4, 7 has a common difference of three. Its fourth term is −2 + (4 − 1) × 3 = 7, and its four-term sum is 4 × (−2 + 7)/2 = 10. The first term occupies position one, so using n differences would move one step too far.
A zero common difference produces a constant sequence; a negative difference produces a descending one. The finite-sum formula still applies in those cases. Check whether an index in a separate exercise starts at zero before copying it into a one-based term-number input.
Algebra & Arithmetic calculators: Geometric Sequence Calculator, Percent Difference Calculator, Proportion Calculator, Roman Numeral Calculator, Completing the Square Calculator, Dot Product Calculator.
An arithmetic sequence adds the same step each time, so its total does not need every term written out. Pair the first term with the last, the second with the second-to-last, and every pair sums to the same figure. Ten terms starting at 3 and stepping by 4 finish at 39 and total 210.
The shortcut is why a savings plan that rises by a fixed deposit each month can be totaled in one line.
Confirm the step is truly constant. If it grows by a percentage instead, you have a geometric sequence and this total will be too low.
Compare with the geometric sequence calculator, where terms multiply, and the Fibonacci page for a recursive pattern. Simple interest on the finance calculators grows arithmetically, while compound interest grows geometrically. The mean and median calculator gives the average of any list.
What is an arithmetic sequence?
A list of numbers where each term differs from the previous one by a constant amount, called the common difference.
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence adds a constant each step, and a geometric sequence multiplies by a constant ratio.
How do I find the sum of an arithmetic sequence?
Multiply the number of terms by the average of the first and last terms. For 10 terms from 3 to 39 that is 10 times 21, or 210.