The arithmetic sequence calculator lets you calculate various important values for an arithmetic sequence. You can calculate the first term, $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$term, common difference, sum of $\hspace{0.2em} n \hspace{0.2em}$ terms, number of terms, or position of a term in the arithmetic sequence.

The calculator will not only give you the answer but also a step-by-step solution.

Here's a quick overview of what an arithmetic sequence is.

## Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant.

Here's an example of an arithmetic sequence.

$5,\hspace{0.35em} 9,\hspace{0.35em} 13,\hspace{0.35em} 17,\hspace{0.35em} 21,\hspace{0.35em} 25, ...$

The constant difference is known as the common difference and is usually denoted by $\hspace{0.2em} d \hspace{0.2em}$. For the sequence above, the common difference would be $\hspace{0.2em} 9 - 5 = 4 \hspace{0.2em}$.

The $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$ term of an arithmetic sequence is denoted by $\hspace{0.2em} a_n \hspace{0.2em}$.

### General Term and the Sum of Terms

For an arithmetic sequence with a first term $\hspace{0.2em} a_1 \hspace{0.2em}$ and common difference $\hspace{0.2em} d \hspace{0.2em}$, the $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$ term is given by —

$a_n = a_1 + (n - 1)d$

And the sum of $\hspace{0.2em} n \hspace{0.2em}$ terms (denoted by $\hspace{0.2em} S_n \hspace{0.2em}$) would be

$S_n = \frac{n}{2} (a_1 + a_n)$

Alternatively, we can use the first equation get rid of $\hspace{0.2em} a_n \hspace{0.2em}$ from the formula for $\hspace{0.2em} S_n \hspace{0.2em}$

$\begin{align*} S_n \hspace{0.3em} &= \hspace{0.3em} \frac{n}{2} \hspace{0.1em} (a_1 + {\color{Red} a_n} ) \\[1.75em] &= \hspace{0.3em} \frac{n}{2} \hspace{0.1em} (a_1 + {\color{Red} a_1 + (n - 1)d} ) \\[1.75em] &= \hspace{0.3em} \frac{n}{2} \hspace{0.1em} (2a_1 + (n - 1)d) \end{align*}$

### Examples

Let's look at a couple of simple examples.

Example

Given the first term $\hspace{0.2em} a_1 = 2 \hspace{0.2em}$ and the common difference $\hspace{0.2em} d = 4 \hspace{0.2em}$, find the $\hspace{0.2em} 15^{\text{th}} \hspace{0.2em}$ term of the arithmetic sequence.

Solution

When we know the first term $\hspace{0.2em} (a_1) \hspace{0.2em}$, common difference $\hspace{0.2em} (d) \hspace{0.2em}$, and $\hspace{0.2em} (n) \hspace{0.2em}$, finding the $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$ term $\hspace{0.2em} (a_n) \hspace{0.2em}$ is quite straightforward. We use the formula,

$a_n \hspace{0.2em} = \hspace{0.2em} a_1 + (n - 1) \cdot d$

Substituting the known values into the formula, we have

$\begin{align*} a_{15} \hspace{0.2em} &= \hspace{0.2em} 2 + (15 - 1) \cdot 4 \\[1em] &= \hspace{0.2em} 58 \end{align*}$

So the $\hspace{0.2em} 15^{\text{th}} \hspace{0.2em}$ term of the given arithmetic sequence is $\hspace{0.2em} 58 \hspace{0.2em}$.

Example

If the sum of the first $\hspace{0.2em} 11 \hspace{0.2em}$ terms of an arithmetic sequence is $\hspace{0.2em} 308 \hspace{0.2em}$, and the first term is $\hspace{0.2em} 3 \hspace{0.2em}$, find the common difference.

Solution

This time we can use the following formula.

$S_n \hspace{0.25em} = \hspace{0.25em} \frac{n}{2} \hspace{0.1em} (2a_1 + (n - 1)d)$

Substituting the values of $\hspace{0.2em} S_n \hspace{0.2em}$, $\hspace{0.2em} a_1 \hspace{0.2em}$, and $\hspace{0.2em} n \hspace{0.2em}$, we get

$308 \hspace{0.25em} = \hspace{0.25em} \frac{11}{2} \cdot (2 \cdot 3 + (11 - 1) \cdot d)$

And solving for $\hspace{0.2em} d \hspace{0.2em}$,

$d \hspace{0.25em} = \hspace{0.25em} 5$