A linear (arithmetic) number sequence models anything that changes by a fixed step per period: a salary rising by the same amount each year, deposits increased monthly, or the number of seats per row. The sum gives the total over all periods.
How to use the Sum of Linear Number Sequence Calculator
- Initial value a
- Increase per period d
- Number of periods n
Results update as you type, and everything is calculated in your browser.
How it works
Enter the initial value, the increase per period and the number of periods.
Inputs explained
| Field | What to enter |
|---|---|
| Initial value a | A number |
| Increase per period d | A number |
| Number of periods n | A number |
What each result means
- Final value = [Initial value a] + ([Number of periods n] – 1)×[Increase per period d]
- Sum of all periods = [Number of periods n]×([Initial value a] + [Final value])÷2
- Average per period = ([Initial value a] + [Final value])÷2
Example values
How the main result changes when Initial value a changes and everything else stays at the example values above:
| Initial value a | Sum of all periods |
|---|---|
| 50 | 1,260 |
| 75 | 1,560 |
| 100 | 1,860 |
| 125 | 2,160 |
| 150 | 2,460 |
| 200 | 3,060 |
Formula
Final value = a + (n − 1)·d, Sum = n·(a + final)/2
Worked example
Inputs — Initial value a: 100; Increase per period d: 10; Number of periods n: 12.
Results — Final value = 210; Sum of all periods = 1,860; Average per period = 155.
Key terms
- Linear sequence
- An arithmetic sequence.
Frequently asked questions
What is a linear sequence?
One that changes by the same amount each step – an arithmetic sequence.
How much do I save if I deposit 100, then 110, 120, … for 12 months?
12 × (100 + 210)/2 = 1,860.
Can the increase be negative?
Yes, for a decreasing sequence.
What is the average term?
The average of the first and last terms.