Arithmetic Sequence Calculator (nth Term and Sum)

Enter the first term (the first number of the sequence), the common difference (how much each term goes up) and the number of terms (how far to go), then press "Calculate". You get the first few terms, the nth term and the sum of the first n terms at once. The formula below is linked to the input fields, so you can also edit the first term, common difference and number of terms right inside the formula.

First term a1

Common difference d

Number of terms n

The first term and the common difference can be decimals or negative numbers. Enter the number of terms n as a positive whole number (1 to 100000).

Result and graph

Enter the first term, the common difference and the number of terms in the fields on the left and press "Calculate". The result and a graph will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and graphs
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter just three values, the first term, the common difference (how much each term goes up) and the number of terms, to get the $n$th term and the sum of the first $n$ terms at once
Works directly for problems like "The sequence 2, 7, 12, 17, … goes up by 5 each time. What is the 20th number? What do the first 20 numbers add up to?"
The first term and the common difference can be decimals or negative numbers (for a decreasing sequence, make the common difference negative)
Along with the result, a graph shows how the terms line up on a straight line (the common difference is how fast they rise)
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

This page works only for arithmetic sequences (sequences where the difference between neighboring terms is always the same). It does not work for sequences that grow by multiplying by the same number, such as 2, 6, 18, … (geometric sequences).

What is this calculation used for?

Estimate the total of a growing savings plan (money planning)

A savings plan of "save $50 the first month, then $25 more each month" is an arithmetic sequence with first term 50 and common difference 25. In month 12 you save $50 + 11 \times 25 = 325$ dollars, and the total for the year is $12 \times (50 + 325) \div 2 = 2250$ dollars. Two formulas let you see far ahead.
Any plan that goes up (or down) by the same amount each time, and the questions "How much is it the kth time?" and "What is the total?", are arithmetic sequence calculations.

Count the seats in a theater or stadium (design and event planning)

Halls with seats that fan out often have rows like "20 seats in row 1, and 2 more seats in each row behind it". With 30 rows, the last row has $20 + 29 \times 2 = 78$ seats, and the whole hall has $30 \times (20 + 78) \div 2 = 1470$ seats.
Without counting row by row, the sum formula quickly gives the total number of seats and the most tickets you can sell.

Add up consecutive numbers in an instant (the story of young Gauss)

"What do you get when you add all the whole numbers from 1 to 100?" The famous story says the mathematician Gauss, as a child, answered at once: $100 \times (1 + 100) \div 2 = 5050$. This is exactly the sum of the arithmetic sequence with first term 1, common difference 1 and 100 terms.
Totals of consecutive numbers, such as adding up the numbers on a run of invoices or stacking "1 on day 1, 2 on day 2, …", are instant with this formula.

Plan a workout or study load that grows little by little (training)

If you plan your runs as "2 miles in week 1, then 0.5 mile more each week", in week 12 you run $2 + 11 \times 0.5 = 7.5$ miles, and the total over 12 weeks is $12 \times (2 + 7.5) \div 2 = 57$ miles.
Before you start, you can check in numbers where a "a little more each time" plan takes you and how much it adds up to.

Simple interest (basic finance)

With simple interest, where interest is paid only on the original amount (the principal), the balance grows by the same amount every year, so it forms an arithmetic sequence. With $10,000 at 3% simple interest per year, it grows by $300 a year, and after 5 years the balance is $10000 + 5 \times 300 = 11500$ dollars.
With compound interest, where interest also earns interest, the balance forms a geometric sequence (a sequence that grows by multiplying). The difference between simple and compound interest is "grow by adding, or grow by multiplying". Seeing it in terms of sequences makes financial products much easier to understand.

Formulas and graphs

Formula for the $n$th term (general term)

Graph

Standard notation (the usual math form)

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

In words (symbols replaced with words)

④ $a_n$: $n$th term $=$ ① $a_1$: first term $+$ ② $(n-1)$: number of steps $\times$ ③ $d$: common difference

The formula in words

① Start with the $a_1$: first term , then add the

② $(n-1)$: number of steps times the

③ $d$: common difference (that is, add the common difference $n-1$ times)

④ and you get the $a_n$: $n$th term

Quick example

For the arithmetic sequence with first term 2 and common difference 5 (2, 7, 12, 17, …), term 20 is

$a_{20}$: term 20 $=$ first term (2) $+$ steps (20 − 1 = 19) $\times$ common difference (5)

$2 + (20 - 1) \times 5 = 2 + 95 = 97$

Key idea

The most common mistake is multiplying by $n$ instead of $n-1$. Going from term 1 to term $n$, you add the common difference once per step, and there are $n-1$ steps, one fewer than $n$. For example, from term 1 to term 3 you add only twice ($3-1$ times). The formula works as is when the common difference $d$ is negative. In that case, the values get smaller with each step.

Formula for the sum of the first $n$ terms

Figure

Standard notation (the usual math form)

$S_n$ $=$ $n$ $\times$ $($ $a_1$ $+$ $a_n$ $)$ $\div$ $2$

In words (symbols replaced with words)

⑤ $S_n$: sum of terms 1 through $n$ $=$ ③ $n$: number of terms $\times$ $($ ① $a_1$: first term $+$ ② $a_n$: $n$th term (last term) $)$ $\div$ ④ $2$: take half

The formula in words

① Add the $a_1$: first term and the

② $a_n$: $n$th term (last term) ,

③ multiply by the $n$: number of terms ,

④ then divide by $2$: take half ,

⑤ and you get the $S_n$: sum of terms 1 through $n$

Quick example

For the arithmetic sequence with first term 2 and common difference 5, the sum of terms 1 through 20 (term 20 is 97) is

$S_{20}$: sum up to term 20 $=$ number of terms (20) $\times$ $($ first term (2) $+$ term 20 (97) $)$ $\div$ take half (2)

$20 \times (2 + 97) = 20 \times 99 = 1980$

$1980 \div 2 = 990$

Key idea

$(a_1 + a_n) \div 2$ is the average of the first term and the last term. An arithmetic sequence goes up by the same amount every step, so the average of all its terms is exactly the average of the first and last terms. That is why the sum is "average × count". This is the whole idea of the formula. Textbooks write it as a fraction, $S_n = \dfrac{n(a_1 + a_n)}{2}$. It is the same formula as above. "What is the sum of all whole numbers from 1 to 100?" is the sum of an arithmetic sequence with first term 1, last term 100 and 100 terms, so the answer is quick: $100 \times (1 + 100) \div 2 = 5050$.

Formula for the sum straight from the first term and common difference

Standard notation (the usual math form)

$S_n$ $=$ $n$ $\times$ $\{$ $2a_1$ $+$ $(n-1)$ $\times$ $d$ $\}$ $\div$ $2$

In words (symbols replaced with words)

⑥ $S_n$: sum of terms 1 through $n$ $=$ ④ $n$: number of terms $\times$ $\{$ ① $2a_1$: twice the first term $+$ ② $(n-1)$: number of steps $\times$ ③ $d$: common difference $\}$ $\div$ ⑤ $2$: take half

The formula in words

① To $2a_1$: twice the first term , add the

② $(n-1)$: number of steps times the

③ $d$: common difference (this equals first term + last term),

④ multiply by the $n$: number of terms ,

⑤ then divide by $2$: take half ,

⑥ and you get the $S_n$: sum of terms 1 through $n$

Quick example

With first term 2, common difference 5 and 20 terms, the sum calculated directly, without finding term 20 first, is

$S_{20}$: sum up to term 20 $=$ number of terms (20) $\times$ $\{$ twice the first term (2 × 2 = 4) $+$ steps (19) $\times$ common difference (5) $\}$ $\div$ take half (2)

$2 \times 2 + (20 - 1) \times 5 = 4 + 95 = 99$

$20 \times 99 \div 2 = 990$

Key idea

This is the first formula (the general term) substituted for $a_n$ in the second formula. Since $a_1 + a_n = a_1 + a_1 + (n-1)d = 2a_1 + (n-1)d$, you can get the sum straight from the first term and the common difference, without finding the last term $a_n$ first. Textbooks write it as a fraction, $S_n = \dfrac{n\{2a_1 + (n-1)d\}}{2}$. Both formulas give the same answer. Use the second formula when you know the last term, and this one when you only know the first term and the common difference.

An arithmetic sequence keeps adding the same number (the common difference). The $n$th term is "the first term plus the common difference added $n-1$ times", and the sum is "(first term + last term) × number of terms ÷ 2", which is the average of the first and last terms times the count.

Symbols and terms

Symbols

$a_1$	a sub one	The first term. The very first term (number) of the sequence. For the sequence 2, 7, 12, 17, …, it is 2.
$d$	dee	The common difference. The difference between neighboring terms (how much each term goes up). For 2, 7, 12, 17, … it is 5. The letter comes from "difference", and it can be a negative number.
$n$	en	The number of terms. A positive whole number that says how far along the sequence you go. "Up to term 20" is $n = 20$.
$a_n$	a sub n	The $n$th term. The $n$th number of the sequence. The small letter at the lower right (the subscript) tells which position it is. The $n$th term written as a formula in $n$ is called the general term.
$n-1$	n minus one	The number of steps from term 1 to term $n$, which is how many times you add the common difference. It is one fewer than the number of terms $n$ (from term 1 to term 3 is 2 steps).
$S_n$	S sub n	The sum of terms 1 through $n$. A short way to write $a_1 + a_2 + \cdots + a_n$. The letter S comes from "sum".
$\cdots$	dot dot dot (ellipsis)	A symbol that says the pattern continues the same way. Writing 2, 7, 12, 17, … says the numbers keep going up by 5.

Terms

sequence	A list of numbers in a set order. Each number in the list is called a term.
arithmetic sequence	A sequence where the difference between neighboring terms is always the same, such as 2, 7, 12, 17, … (the difference is always 5). It is also called an arithmetic progression.
term	Each single number in a sequence. From the start, they are called term 1, term 2, and so on.
first term	The very first term of a sequence, that is, term 1. It is written $a_1$.
common difference	In an arithmetic sequence, the difference between neighboring terms. You get the same value whichever neighboring pair you subtract. It can be negative (for a decreasing sequence) or a decimal.
last term	The very last term in the range you are looking at. For the sum of terms 1 through $n$, the last term is $a_n$.
general term	The $n$th term written as a formula in $n$, also called the explicit formula. For an arithmetic sequence it is $a_n = a_1 + (n-1)d$. Put a position number in for $n$ and you get that term right away.
geometric sequence	A sequence where the ratio between neighboring terms is always the same, such as 2, 6, 18, 54, … (multiply by 3 each time). It grows differently from an arithmetic sequence, which adds the same number, so the formulas on this page do not apply.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to review these topics is the fastest way forward.

Number patterns (Grades 4–5)	Being able to spot a rule such as "it goes up by 5 each time" in a list of numbers like 2, 7, 12, 17, … Being able to make a table that pairs each position number with its value
Negative numbers (Grade 7)	Being able to add a negative number, as in $10 + (-2) = 8$ Being able to multiply by a negative number, as in $(6-1) \times (-2) = -10$
Variables and expressions (Grade 6)	Knowing that letters such as $n$ and $a_1$ stand for numbers that can change Being able to substitute, such as putting $n = 20$ into $n - 1$ to get 19
Order of operations (Grade 5)	Doing multiplication before addition (in $2 + 19 \times 5$, do $19 \times 5$ first) Doing what is inside parentheses first (in $20 \times (2 + 97)$, do $2 + 97$ first)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the nth term (first term 2, common difference 5, 20 terms)

First term a1	2
Common difference d	5
Number of terms n	20
nth term	=B1+(B3-1)*B2

Table to find the sum of terms 1 through n (using the nth term)

First term a1	2
Common difference d	5
Number of terms n	20
nth term	=B1+(B3-1)*B2
Sum of terms 1 through n	=B3*(B1+B4)/2

Table to find the sum straight from the first term and common difference

First term a1	2
Common difference d	5
Number of terms n	20
Sum of terms 1 through n	=B3(2B1+(B3-1)*B2)/2

After pasting, cells B1 to B3 are the inputs and the bottom cell shows the calculated result.
In the formulas, "B1" and "B3" say "use the number in that cell", "*" is multiplication and "/" is division. "=B1+(B3-1)*B2" is the formula itself: "first term plus (number of terms − 1) × common difference".
For example, the first table shows 97 (term 20) in B4, and the second table shows 990 (the sum) in B5. Just replace B1 to B3 with your own numbers.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the nth term (first term 2, common difference 5, 20 terms)

First term a1	2
Common difference d	5
Number of terms n	20
nth term	=B1+(B3-1)*B2

Table to find the sum of terms 1 through n (using the nth term)

First term a1	2
Common difference d	5
Number of terms n	20
nth term	=B1+(B3-1)*B2
Sum of terms 1 through n	=B3*(B1+B4)/2

Table to find the sum straight from the first term and common difference

First term a1	2
Common difference d	5
Number of terms n	20
Sum of terms 1 through n	=B3(2B1+(B3-1)*B2)/2

The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace B1 to B3 with your own numbers.

How to calculate it in Python

first_term = 2         # first term
common_difference = 5  # common difference
number_of_terms = 20   # number of terms

# nth term = first term + (n - 1) * common difference
nth_term = first_term + (number_of_terms - 1) * common_difference
# sum of terms 1 through n = n * (first term + nth term) / 2
sum_of_terms = number_of_terms * (first_term + nth_term) / 2
# first 10 terms of the sequence (for checking)
sequence = [first_term + k * common_difference for k in range(min(number_of_terms, 10))]

print(f"First terms: {sequence}")
print(f"Term {number_of_terms}: {nth_term}")
print(f"Sum of terms 1 through {number_of_terms}: {sum_of_terms}")

It runs with the standard library only. Change the three numbers at the top (first term, common difference, number of terms) and run it. This example prints 97 for term 20 and 990.0 for the sum. To show the sum as a whole number, change "/ 2" to "// 2" (integer division) when the first term and common difference are whole numbers.

How to write it in LaTeX and other math languages (copy and paste)

Formula for the $n$th term (general term)

aₙ = a₁ + (n − 1)d

a_n = a_1 + (n-1)d

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>a</mi><mi>n</mi></msub>
    <mo>=</mo>
    <msub><mi>a</mi><mn>1</mn></msub>
    <mo>+</mo>
    <mrow><mo>(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo>)</mo></mrow>
    <mi>d</mi>
  </mrow>
</math>

a_n = a_1 + (n-1) d

a1 + (n - 1)*d

an := a1 + (n - 1)*d;

a_n = a1 + (n - 1)*d;

a_n = a_1 + (n - 1)d

Formula for the sum of the first $n$ terms

Sₙ = n(a₁ + aₙ) ÷ 2

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

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>S</mi><mi>n</mi></msub>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <mi>n</mi>
        <mo>(</mo>
        <msub><mi>a</mi><mn>1</mn></msub>
        <mo>+</mo>
        <msub><mi>a</mi><mi>n</mi></msub>
        <mo>)</mo>
      </mrow>
      <mn>2</mn>
    </mfrac>
  </mrow>
</math>

S_n = (n(a_1 + a_n))/2

n*(a1 + an)/2

Sn := n*(a1 + an)/2;

S_n = n*(a1 + a_n)/2;

S_n = n(a_1 + a_n)/2

Formula for the sum straight from the first term and common difference

Sₙ = n{2a₁ + (n − 1)d} ÷ 2

S_n = \dfrac{n\{2a_1 + (n-1)d\}}{2}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>S</mi><mi>n</mi></msub>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <mi>n</mi>
        <mo>{</mo>
        <mn>2</mn>
        <msub><mi>a</mi><mn>1</mn></msub>
        <mo>+</mo>
        <mrow><mo>(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo>)</mo></mrow>
        <mi>d</mi>
        <mo>}</mo>
      </mrow>
      <mn>2</mn>
    </mfrac>
  </mrow>
</math>

S_n = (n(2 a_1 + (n-1) d))/2

n*(2*a1 + (n - 1)*d)/2

Sn := n*(2*a1 + (n - 1)*d)/2;

S_n = n*(2*a1 + (n - 1)*d)/2;

S_n = n(2a_1 + (n - 1)d)/2

How to have ChatGPT do the calculation

You are an assistant for arithmetic sequence calculations. Do the following calculations by actually running Python code, and base your answer only on the numbers from the output (do not answer from mental math or guesses). Find the nth term with a_n = a_1 + (n-1)d and the sum with S_n = n(a_1 + a_n)/2.

1. Term 20 and the sum of terms 1 through 20 of the arithmetic sequence with first term 2 and common difference 5
2. Term 10 and the sum of terms 1 through 10 of the arithmetic sequence with first term -3 and common difference 4
3. Term 15 and the sum of terms 1 through 15 of the arithmetic sequence with first term 100 and common difference -7

Show the formulas you used and the numbers from the output.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula

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\(a_1\)	a sub one	The first term. The very first term (number) of the sequence. For the sequence 2, 7, 12, 17, …, it is 2.
\(d\)	dee	The common difference. The difference between neighboring terms (how much each term goes up). For 2, 7, 12, 17, … it is 5. The letter comes from "difference", and it can be a negative number.
\(n\)	en	The number of terms. A positive whole number that says how far along the sequence you go. "Up to term 20" is \(n = 20\).
\(a_n\)	a sub n	The \(n\)th term. The \(n\)th number of the sequence. The small letter at the lower right (the subscript) tells which position it is. The \(n\)th term written as a formula in \(n\) is called the general term.
\(n-1\)	n minus one	The number of steps from term 1 to term \(n\), which is how many times you add the common difference. It is one fewer than the number of terms \(n\) (from term 1 to term 3 is 2 steps).
\(S_n\)	S sub n	The sum of terms 1 through \(n\). A short way to write \(a_1 + a_2 + \cdots + a_n\). The letter S comes from "sum".
\(\cdots\)	dot dot dot (ellipsis)	A symbol that says the pattern continues the same way. Writing 2, 7, 12, 17, … says the numbers keep going up by 5.

Arithmetic Sequence Calculator (nth Term and Sum)

Table of Contents

What you can do on this page

What is this calculation used for?

Formulas and graphs

Symbols and terms

Symbols

Terms

Good to know before you start

How to calculate it in Excel

How to calculate it in Google Sheets

How to calculate it in Python

How to write it in LaTeX and other math languages (copy and paste)

How to have ChatGPT do the calculation

How to Use

DataChef Features

Related Features

Geometric Sequence Calculator (nth Term and Sum)

Finds the nth term and the sum from the first term, common ratio and number of terms (a sequence that grows by multiplying by the same number, the partner of the arithmetic sequence, which grows by adding)

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Finds the slope, distance, angle and equation of a line from two points (on a graph, the common difference of an arithmetic sequence is the slope of a line)

Average Calculator (Mean)

Finds the average (arithmetic mean), median and more from a list of numbers (the sum of an arithmetic sequence is the average of the first and last terms times the number of terms)

Fibonacci Sequence Calculator (nth Term and Sum)

Finds the nth term and the sum of the Fibonacci sequence (1, 1, 2, 3, 5, 8, …), a sequence that adds the two previous terms instead of a fixed number

Recurrence Relation Calculator (General Term with Steps)

Solves the recurrence a(n+1) = p·a(n) + q for a closed-form formula, with steps using the fixed point (the case p = 1 is an arithmetic sequence)

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