Arithmetic Sequence Calculator

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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, nth\hspace{0.2em} n^{\text{th}} \hspace{0.2em}term, common difference, sum of n\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

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i. Valid Inputs

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

  • Whole numbers or decimals → 2\hspace{0.2em} 2 \hspace{0.2em}, 4.25\hspace{0.2em} -4.25 \hspace{0.2em}, 0\hspace{0.2em} 0 \hspace{0.2em}, 0.33\hspace{0.2em} 0.33 \hspace{0.2em}
  • Fractions → 2/3\hspace{0.2em} 2/3 \hspace{0.2em}, 1/5\hspace{0.2em} -1/5 \hspace{0.2em}
  • Mixed numbers → 51/4\hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em}

Number of terms n\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.

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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.

5,9,13,17,21,25,...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 d\hspace{0.2em} d \hspace{0.2em}. For the sequence above, the common difference would be 95=4\hspace{0.2em} 9 - 5 = 4 \hspace{0.2em}.

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

General Term and the Sum of Terms

For an arithmetic sequence with a first term a1\hspace{0.2em} a_1 \hspace{0.2em} and common difference d\hspace{0.2em} d \hspace{0.2em}, the nth\hspace{0.2em} n^{\text{th}} \hspace{0.2em} term is given by —

an=a1+(n1)da_n = a_1 + (n - 1)d

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

Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)

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

Sn=n2(a1+an)=n2(a1+a1+(n1)d)=n2(2a1+(n1)d)\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*}


Let's look at a couple of simple examples.


Given the first term a1=2\hspace{0.2em} a_1 = 2 \hspace{0.2em} and the common difference d=4\hspace{0.2em} d = 4 \hspace{0.2em}, find the 15th\hspace{0.2em} 15^{\text{th}} \hspace{0.2em} term of the arithmetic sequence.


When we know the first term (a1)\hspace{0.2em} (a_1) \hspace{0.2em}, common difference (d)\hspace{0.2em} (d) \hspace{0.2em}, and (n)\hspace{0.2em} (n) \hspace{0.2em}, finding the nth\hspace{0.2em} n^{\text{th}} \hspace{0.2em} term (an)\hspace{0.2em} (a_n) \hspace{0.2em} is quite straightforward. We use the formula,

an=a1+(n1)da_n \hspace{0.2em} = \hspace{0.2em} a_1 + (n - 1) \cdot d

Substituting the known values into the formula, we have

a15=2+(151)4=58\begin{align*} a_{15} \hspace{0.2em} &= \hspace{0.2em} 2 + (15 - 1) \cdot 4 \\[1em] &= \hspace{0.2em} 58 \end{align*}

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


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


This time we can use the following formula.

Sn=n2(2a1+(n1)d)S_n \hspace{0.25em} = \hspace{0.25em} \frac{n}{2} \hspace{0.1em} (2a_1 + (n - 1)d)

Substituting the values of Sn\hspace{0.2em} S_n \hspace{0.2em}, a1\hspace{0.2em} a_1 \hspace{0.2em}, and n\hspace{0.2em} n \hspace{0.2em}, we get

308=112(23+(111)d)308 \hspace{0.25em} = \hspace{0.25em} \frac{11}{2} \cdot (2 \cdot 3 + (11 - 1) \cdot d)

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

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

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