How do you find the nth term of an arithmetic sequence?

Multiply the difference d by (n-1) and add it to the first term a_1. Symbolically, this is expressed as: a_n = a_1 + (n - 1)d. For example, if the first term is 1 and the difference is 2, then the fifth term is equal to 1 + 4 times 2, which is 9.

How do you calculate the difference?

Subtract any term from the immediately following term. It can be any adjacent pair, for example the second term minus the first term. If different pairs of terms result in different differences then the sequence is not arithmetic.

What is the difference between a sequence and a series?

A sequence is an ordered collection of elements, while a series is the sum of those elements. The sum of the first n terms is called a partial sum and can be calculated using the formula S_n = n * (a_1 + a_n) / 2.

What is the difference between an arithmetic sequence and a geometric sequence?

In an arithmetic sequence the difference between consecutive terms is constant so you add as you go along the sequence. In a geometric sequence the ratio is constant so you multiply as you go along the sequence.

Can this calculator determine the first term or number of terms?

Yes. Enter any three of the following values: first term, difference, position or nth term and leave the desired term blank. The calculator will transform the formula for the nth term and provide the missing value.

Arithmetic Sequence Calculator

Your sequence

First term, a₁

Common difference, d

Term position / number of terms, n

Value of the nth term, aₙ

Sum a chosen range of terms

Add the terms from a starting index up to n, instead of from the first term.

Enter any three of the first term, common difference, term position, and nth term. The calculator solves for the fourth.

Please input values into the fields

Chart and table

An arithmetic sequence is a series of numbers in which each number is a constant difference from the preceding number. This constant difference is called the common difference. If you know the first term and the common difference, you can find any term in the sequence by using the formula for finding the nth term.

This calculator allows you to find the unknown member of an arithmetic sequence given any other relevant known value, as well as calculate the sum of the sequence up to a certain limit and represent it graphically. Enter three values from this set: first term, difference between terms, position of the term in the sequence or the actual value of the term.

What is an Arithmetic Sequence?

A sequence is an ordered list of numbers in which each number is called a term. If the difference between two consecutive terms is always the same then it is called an arithmetic sequence. This common difference is denoted by d.

The sequence of numbers 3, 5, 7, 9, 11 is an arithmetic progression since each element increases by 2, so d = 2. The difference need not be a whole number and can also be negative. If d is positive the sequence will increase in value, if d is negative it will decrease in value, and if d is zero then all elements of the sequence are equal.

The formula for the nth element:

To find the element at position n, start with the first element and add the common difference multiplied by (n-1).

a_n = a_1 + (n - 1)\,d

In this formula, a1 is the first element in the sequence, d is the difference between consecutive elements, and n is the position of the desired element. If we consider the sequence 1, 3, 5, 7, 9... where a1 = 1 and d = 2, we want to find the fifth element in the sequence.

a_5 = 1 + (5 - 1)\times 2 = 1 + 8 = 9

The fifth term is 9 and matches the sequence written. Computational tools use the same formula but in reverse order. So regardless of which values are known (first term, difference, position or the term itself), one can calculate the missing value by simply leaving a blank space.

Sum of Sequences and Series

A sequence is a list of numbers, while a series is the result of adding those numbers together. The sum of the first n elements is usually referred to as a partial sum and denoted by Sn. There's a short formula for it:

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

It can be considered as the product of the number of elements and the average of the first and last element. If the last element is unknown, the formula for the nth term can be used to substitute it in, leaving only the first element and the difference required.

S_n = \frac{n}{2}\,\bigl(2a_1 + (n - 1)\,d\bigr)

The sum of the first five elements in the sequence 1, 3, 5, 7, 9 is as follows:

S_5 = \frac{5}{2}\,(1 + 9) = \frac{5}{2}\times 10 = 25

Calculate the difference

To determine the difference of a given sequence, subtract any element from the element that immediately follows it. In a true arithmetic sequence, each pair of consecutive elements will yield the same result. If the differences between individual elements do not match up, then you are not working with an arithmetic sequence.

d = a_{n} - a_{n-1}

Instructions for using this calculator

Start with three known values. To find a particular element, enter the first element, the difference and the position leaving the nth term blank. Conversely you can enter the value of an element and leave the position blank to discover where it is located. Alternatively you can leave the difference blank to reconstruct the increments from known elements.

The sum of the first n elements is shown along with each element. The middle element is halfway between the first and last element. If the Range option is selected, only the elements from the selected starting position up to n are calculated. Selecting Graph and Table allows you to view the full sequence of numbers.

Symbol overview

Symbol	Meaning	Example
a_1	First term	1
d	Common difference	2
n	Term position or number of terms	5
a_n	Value of the nth term	9
S_n	Sum of the first n terms	25

Arithmetic and geometric sequences

An arithmetic sequence increases by adding a constant and can therefore be represented as a straight line. A geometric sequence increases by multiplying by a constant factor and either goes up or approaches zero gradually. If the difference between consecutive terms is constant, it's an arithmetic sequence. If the result of dividing is constant, it's a geometric sequence.

Application examples for arithmetic sequences

Arithmetic sequences can be used to represent anything that varies in equal steps. Examples include the number of seats in each row of a theater, which increases by a constant amount; salary increments with an annual raise of the same size; monthly savings deposits; and markers on surveying lines - all follow an arithmetic pattern. The nth term allows you to go directly to any position, and the sum allows you to calculate the total at once.

This calculator is for learning and planning purposes only. If you are using it for homework, please check with your class what the first term is defined as. This is because some textbooks start indexing at zero.

Frequently asked questions

How do you find the nth term of an arithmetic sequence?: Multiply the difference d by (n-1) and add it to the first term a_1. Symbolically, this is expressed as: a_n = a_1 + (n - 1)d. For example, if the first term is 1 and the difference is 2, then the fifth term is equal to 1 + 4 times 2, which is 9.
How do you calculate the difference?: Subtract any term from the immediately following term. It can be any adjacent pair, for example the second term minus the first term. If different pairs of terms result in different differences then the sequence is not arithmetic.
What is the difference between a sequence and a series?: A sequence is an ordered collection of elements, while a series is the sum of those elements. The sum of the first n terms is called a partial sum and can be calculated using the formula S_n = n * (a_1 + a_n) / 2.
What is the difference between an arithmetic sequence and a geometric sequence?: In an arithmetic sequence the difference between consecutive terms is constant so you add as you go along the sequence. In a geometric sequence the ratio is constant so you multiply as you go along the sequence.
Can this calculator determine the first term or number of terms?: Yes. Enter any three of the following values: first term, difference, position or nth term and leave the desired term blank. The calculator will transform the formula for the nth term and provide the missing value.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Arithmetic progression
Definition of an arithmetic progression, the nth term, and the sum formula.
Wolfram MathWorld: Arithmetic Progression
Reference treatment of arithmetic progressions and partial sums.