# Sum of a Linear Number Sequence Calculator

This calculator deals with finding the sum of a linear number sequence - a set of values where every consecutive one differs by the same amount from the previous one. An example would be 20, 23, 26, 29 - it's a sequence of `4`

numbers, with `20`

as the initial value, `29`

as the final value, and `3`

as the difference between the successive figures. The sum of all of these numbers is `98`

.

If you need to quickly find the value of a particular number in a linear number sequence, check out the arithmetic sequence calculator.

## What is the formula for the sum of a linear sequence?

The formula for the sum of a linear sequence reads:

`sum = n / 2 × (2 × a + diff × (n - 1))`

where:

`a`

is the**initial value**of the sequence;`diff`

is the**difference**between any two consecutive numbers is your sequence;`n`

is the number of**periods**(of elements in your sequence); and`sum`

is the**sum**you're looking for.

Let's translate the above formula in a sequence (pun unintended) of easily executable steps!

## How do I find the sum of a linear sequence?

To compute the sum of a linear sequence:

- Write down the
**initial value**of your sequence`a`

. - Compute the
**final value**of your sequence as`a + diff × (n - 1)`

, where`diff`

is the difference of your sequence, and`n`

is the number of its elements. **Add**together the results of Step 1 and Step 2.**Multiply**the result by`n`

.**Divide**the result by`2`

.**That's it!**

## How do I find the final value of a linear sequence?

To determine the final value of a linear sequence:

- Write down the
**initial value**`a`

of your sequence as well as the**difference**`diff`

between the two consecutive steps. - Decide
**how many elements**you want in your sequence. Denote this value by`n`

. - Compute
`diff × (n - 1)`

. - Add
`a`

to the number obtained in Step 3. This is**your result**: the value of the final, i.e., the nth element of your linear sequence.

## A practical example.

Let's say we're selling storage space for photos of goats on the cloud. Customers pay us an amount of money that's directly proportional to the amount of data they upload - `$1 per 1 Gigabyte`

. Alice initially uploads `5GB`

of family photos and then adds an additional `2GB`

per month. For the purpose of our analytics, we want to know how much she stores after a year and how much her business is worth to us during that time. Below, you'll find a table with all of the values and a pre-filled calculator widget.

Month | Payment | Sum |
---|---|---|

1 | 5 | 5 |

2 | 7 | 12 |

3 | 9 | 21 |

4 | 11 | 32 |

5 | 13 | 45 |

6 | 15 | 60 |

7 | 17 | 77 |

8 | 19 | 96 |

9 | 21 | 117 |

10 | 23 | 140 |

11 | 25 | 165 |

12 | 27 | 192 |

## FAQ

### Are linear sequences and arithmetic sequences the same?

**Yes**, the terms *linear sequence* and *arithmetic sequence* describe the same type of sequences: those that arise by always adding the same amount to the previous value to get the next value. This amount is called the *difference* of the linear/arithmetic sequence.

### What is the sum of the first 100 numbers?

The answer is **5050**. This is because the natural numbers form a linear sequence of `n = 100`

elements with the initial value `a = 1`

and the difference of `diff = 1`

. Applying the formula for the sum of linear sequence, we get `sum = n / 2 × (2 × a + diff × (n - 1)) = 100 / 2 × (2 × 1 + 1 × (100 - 1))`

. It follows easily that `sum = 5050`

.