Arithmetic Sequence Calculator

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First Term

\hspace{0.2em} \,(a_1) \hspace{0.2em}

Common Difference

\hspace{0.2em} \,(d) \hspace{0.2em}

Number of Terms

\hspace{0.2em} \,(n) \hspace{0.2em}

Please provide your input and click the calculate button

About the Arithmetic Sequence Calculator

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.

Usage Guide

Hide

i. Valid Inputs

The different inputs (except $\hspace{0.2em} n \hspace{0.2em}$ ) can be a real number in any format as shown below.

Whole numbers or decimals → $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} -4.25 \hspace{0.2em}$ , $\hspace{0.2em} 0 \hspace{0.2em}$ , $\hspace{0.2em} 0.33 \hspace{0.2em}$
Fractions → $\hspace{0.2em} 2/3 \hspace{0.2em}$ , $\hspace{0.2em} -1/5 \hspace{0.2em}$
Mixed numbers → $\hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em}$

Number of terms $\hspace{0.2em} n \hspace{0.2em}$ must be a positive integer (a counting number).

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

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