Sum of Linear Number Sequence Calculator

What Is a Linear Number Sequence?

A linear number sequence (also called an arithmetic sequence) is a list of numbers where each term differs from the previous one by a constant amount called the common difference $d$. If the first term is $a$, the sequence is:

$$a,\; a+d,\; a+2d,\; a+3d,\; \ldots$$

Sum of an Arithmetic Sequence

The sum of the first $n$ terms is:

$$\text{sum} = \frac{n}{2}\left(2a + d(n-1)\right)$$

An equivalent form uses the first and last terms:

$$\text{sum} = \frac{n}{2}(a + a_n)$$

where $a_n = a + d(n-1)$ is the $n$th term.

Example: Sum of 1 Through 100

Here $a = 1$, $d = 1$, and $n = 100$. The sum is:

$$\frac{100}{2}(1 + 100) = 50 \times 101 = 5050$$

Related: Arithmetic Sequence Generator and Number Sequence Calculator.