Answer first
What this calculator tells you
Find the last term and the sum of an arithmetic sequence from its first term, common difference and length. Total a series that steps by a fixed amount, such as a savings ladder or a seating plan by row. Formula: nth term = a₁ + (n−1)d; sum of n terms = n × (a₁ + aₙ) ÷ 2. At the worked-example inputs, the last term (aₙ) is 39. Holding every other input steady, moving first term (a₁) from 1 to 5 moves the result from 37 to 41.
Transparent method
The formula
Total a series that steps by a fixed amount, such as a savings ladder or a seating plan by row.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the last term (aₙ) is 39. It rises with common difference (d), number of terms (n) and first term (a₁).
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects.
Before you rely on it
What to check
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.
The common error
Where people go wrong with arithmetic sequence calculator
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.
Sensitivity evidence
How first term (a₁) changes the last term (aₙ)
Holding every other input at the worked-example value, moving first term (a₁) from 1 to 5 moves the last term (aₙ) from 37 to 41: a spread of 4, or 10% of the worked-example result.
| First term (a₁) | Last term (aₙ) | Sum of all terms |
|---|---|---|
| 1 | 37 | 190 |
| 2 | 38 | 200 |
| 3worked example | 39 | 210 |
| 4 | 40 | 220 |
| 5 | 41 | 230 |
Every input, tested
Which input moves the last term (aₙ) most
Of the 3 inputs, common difference (d) moves the last term (aₙ) most (18 across the range tested) and first term (a₁) moves it least (2).
| Input | Tested from | To | Last term (aₙ) at each end | Swing |
|---|---|---|---|---|
| Common difference (d) | 3 | 5 | 30 to 48 | 18 (46%) |
| Number of terms (n) | 9 | 11 | 35 to 43 | 8 (21%) |
| First term (a₁) | 2 | 4 | 38 to 40 | 2 (5.1%) |
Two variables at once
Last term (aₙ) by first term (a₁) and common difference (d)
Across the grid the last term (aₙ) runs from 19 to 59. Moving first term (a₁) from 1 to 5 shifts it by 4 at the middle column, and moving common difference (d) from 2 to 6 shifts it by 36 at the middle row, so common difference (d) is the bigger lever here.
| 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 |
The highlighted cell is the worked example: 39.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| First term (a₁) | 3 | The starting value of the sequence. |
| Common difference (d) | 4 | The fixed amount added at every step. Negative for a falling sequence. |
| Number of terms (n) | 10 | How many terms to include, from 1 to 1,000. |
| Last term (aₙ) | 39 | |
| Sum of all terms | 210 | |
Inputs, definitions and assumptions
First term (a₁)
The starting value of the sequence. The prefilled worked-example value is 3.
Common difference (d)
The fixed amount added at every step. Negative for a falling sequence. The prefilled worked-example value is 4.
Number of terms (n)
How many terms to include, from 1 to 1,000. The prefilled worked-example value is 10.
How to use this calculator
- 1Verify the inputs. Gather first term (a₁), common difference (d) and number of terms (n) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the last term (aₙ) at 39. Store your own version of it as Scenario A.
- 3Test one change. Start with common difference (d), the input with the biggest effect here: moving common difference (d) from 3 to 5 takes the last term (aₙ) from 30 to 48, a swing of 46% of the worked-example figure.
- 4Check the extremes. At half the example common difference (d) (2) the last term (aₙ) is 21; at double (8) it is 75.
People also ask
Frequently asked questions
How do you calculate arithmetic sequence?
nth term = a₁ + (n−1)d; sum of n terms = n × (a₁ + aₙ) ÷ 2. At the worked-example inputs the last term (aₙ) is 39.
What does the arithmetic sequence result mean?
Total a series that steps by a fixed amount, such as a savings ladder or a seating plan by row. At the worked-example inputs the last term (aₙ) is 39. It rises with common difference (d), number of terms (n) and first term (a₁).
How much does first term (a₁) change the last term (aₙ)?
Holding every other input at the worked-example value, moving first term (a₁) from 1 to 5 moves the last term (aₙ) from 37 to 41, a spread of 4.
What are the limits of this arithmetic sequence calculator?
These are exact mathematical formulas; results are limited only by floating-point precision, not by real-world estimation. Confirm the convention (rounding rule, sign, base) your assignment or application expects. The tables on this page test first term (a₁) only from 1 to 5; a value outside that range is not tabulated here.
Which input moves the last term (aₙ) most in the arithmetic sequence calculator?
Ranked by how far each moves the last term (aₙ) across the range tested: common difference (d) (18, 46%), number of terms (n) (8, 21%) and first term (a₁) (2, 5.1%).
If I double common difference (d) in the arithmetic sequence calculator, does the last term (aₙ) double?
Doubling it from 4 to 8 takes the last term (aₙ) from 39 to 75, which is 1.92 times the worked-example figure. So it grows, but by less than double. Halving it to 2 gives 21.
How much does common difference (d) matter in the arithmetic sequence calculator?
The worked example uses 4. Holding every other input at its worked-example value, moving common difference (d) from 3 to 5 takes the last term (aₙ) from 30 to 48, a swing of 46% of the worked-example figure.
How much does number of terms (n) matter in the arithmetic sequence calculator?
The worked example uses 10. Holding every other input at its worked-example value, moving number of terms (n) from 9 to 11 takes the last term (aₙ) from 35 to 43, a swing of 21% of the worked-example figure.
Which inputs change the sum of all terms in the arithmetic sequence calculator?
At the worked-example inputs it is 210. First term (a₁) takes it from 200 to 220, common difference (d) takes it from 165 to 255 and number of terms (n) takes it from 171 to 253.
Is rounding the same as truncating?
No. Rounding looks at the digit after the cutoff and rounds up or down to the nearest value. Truncating simply cuts off everything after the cutoff regardless of its value. Truncating 2.99 to one decimal gives 2.9. Rounding gives 3.0.
Sources and evidence
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