# Sum of Linear Number Sequence Calculator

URL: https://speedycalculator.com/calculators/math/sum-of-linear-number-sequence/
Last updated: 2026-10-30

> The sum of linear number sequence calculator adds up a sequence that grows by a fixed amount each period – like savings increased by the same amount every month – giving the final value and the total.

## Inputs

- Initial value a
- Increase per period d
- Number of periods n

## Results

- Final value
- Sum of all periods
- Average per period

## Formula

```
Final value = a + (n − 1)·d,  Sum = n·(a + final)/2
```

## Explanation

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

FieldWhat to enterInitial value aA numberIncrease per period dA numberNumber of periods nA 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 aSum of all periods501,260751,5601001,8601252,1601502,4602003,060

## 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.

## FAQ

**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.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)