Geometric Sequence Calculator (nth Term and Sum)

Enter the first term (the first number of the sequence), the common ratio (the number you multiply by at each step) 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 ratio and number of terms right inside the formula. When the absolute value of the common ratio is less than 1, it also shows the sum when you keep adding forever (the sum of the infinite geometric series).

First term a

Common ratio r

Number of terms n

The first term and the common ratio can be decimals or negative numbers. Enter the number of terms n as a positive whole number (1 to 1000).

Result and graph

Enter the first term, the common ratio 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 ratio (the number you multiply by at each step) 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 3, 6, 12, 24, … doubles each time. What is the 10th number? What do the first 10 numbers add up to?"
The first term and the common ratio can be decimals or negative numbers (use a common ratio of "0.5" for a sequence that halves each time, and a negative ratio for one whose signs alternate)
When the absolute value of the common ratio is less than 1, it also shows the sum when you keep adding forever (the sum of the infinite geometric series)
Along with the result, a graph shows how the terms keep doubling (rising along a curve). A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page

This page works only for geometric sequences (sequences where the ratio between neighboring terms is always the same, so you multiply by the same number each step). For sequences that grow by adding the same number, such as 5, 8, 11, … (arithmetic sequences), please use the arithmetic sequence calculator.

What is this calculation used for?

See how money grows with compound interest (money planning)

With compound interest, where interest also earns interest, the balance grows by the same factor every year, so it forms a geometric sequence. If you invest $10,000 at 3% interest compounded yearly, the balance is multiplied by 1.03 every year. After 10 years it is $10000 \times 1.03^{10} \approx 13439$ dollars (assuming the rate stays the same for all 10 years).
"How much will it be in k years?" is exactly the $n$th-term calculation of a geometric sequence. The difference from simple interest, which grows by the same amount every year (an arithmetic sequence), is "grow by adding, or grow by multiplying".

Estimate how bacteria or cells multiply (biology and food safety)

Bacteria that multiply by splitting double in number with each division, so they form a geometric sequence with common ratio 2. If one bacterium divides once every 20 minutes, after 6 hours (18 divisions) there are $2^{18} = 262144$ of them, more than 260,000 (this assumes an ideal case with enough food and space and no bacteria dying).
This doubling pattern (exponential growth) is also the reason you are told not to leave food out at room temperature.

Folding paper doubles its thickness (a feel for exponential growth)

Fold a sheet of copy paper about 0.004 inch thick once and it becomes twice as thick. Each fold doubles it, so the thickness forms a geometric sequence with common ratio 2. On paper, 10 folds give $0.004 \times 2^{10} \approx 4.1$ inches, and 23 folds give about 2,800 feet, taller than the Burj Khalifa (2,717 feet), the tallest building in the world.
In real life you can only fold paper about 7 or 8 times. Still, "just 23 doublings to pass the world's tallest building" shows how far geometric growth goes beyond our feel for addition.

Calculate how a bouncing ball loses height (physics and sports)

Suppose a ball bounces back to 60% of the height it fell from each time. Drop it from 6 feet, and the bounce heights shrink as a geometric sequence with common ratio 0.6. By the 5th bounce, it rises only $6 \times 0.6^{5} \approx 0.47$ feet (about 5.6 inches).
Things that shrink by the same percentage each time, such as fading vibrations and sounds or the level of a medicine in the blood, are modeled with geometric sequences whose common ratio is less than 1.

Musical scales are geometric sequences of frequencies (music and instruments)

The equal-tempered scale used on the piano and other instruments is a geometric sequence: each half step up multiplies the frequency by about 1.0595 (exactly the 12th root of 2). Going up 12 half steps from A (440 Hz) gives $440 \times (\sqrt[12]{2})^{12} = 880$ Hz, the A exactly one octave higher.
Because every half step has the same ratio, music sounds the same in any key you move it to. Geometric sequences are built into the foundation of music.

Formulas and graphs

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

Graph

Standard notation (the usual math form)

$a_n$ $=$ $a$ $\times$ $r$ $n-1$

In words (symbols replaced with words)

④ $a_n$: $n$th term $=$ ① $a$: first term $\times$ ② $r$: common ratio ③ $(n-1)$: number of steps

The formula in words

① Start with the $a$: first term and multiply by the

② $r$: common ratio again and again, as many times as the

③ $(n-1)$: number of steps ,

④ and you get the $a_n$: $n$th term

Quick example

For the geometric sequence with first term 3 and common ratio 2 (3, 6, 12, 24, …), term 5 is

$a_5$: term 5 $=$ first term (3) $\times$ common ratio (2) steps (5 − 1 = 4)

$3 \times 2^{4} = 3 \times 16 = 48$

Key idea

The most common mistake is multiplying $n$ times instead of $n-1$ times. Going from term 1 to term $n$, you multiply by the common ratio once per step, and there are $n-1$ steps, one fewer than $n$. For example, from term 1 to term 3 you multiply only twice ($3-1$ times). If the common ratio $r$ is a positive number less than $1$ (for example 0.5), the terms keep getting smaller and approach $0$. If it is negative (for example $-2$), the signs of the terms alternate. In both cases the formula works as is.

Formula for the sum of the first $n$ terms (when the common ratio is not $1$)

Figure

Standard notation (the usual math form)

$S_n$ $=$ $a$ $\times$ $(1 - r^{n})$ $\div$ $(1 - r)$

In words (symbols replaced with words)

④ $S_n$: sum of terms 1 through $n$ $=$ ① $a$: first term $\times$ ② $(1 - r^n)$: 1 minus the ratio to the $n$th power $\div$ ③ $(1 - r)$: 1 minus the ratio

The formula in words

① Multiply the $a$: first term by

② $(1 - r^n)$: 1 minus the ratio to the $n$th power ,

③ then divide by $(1 - r)$: 1 minus the ratio ,

④ and you get the $S_n$: sum of terms 1 through $n$

Quick example

For the geometric sequence with first term 3 and common ratio 2 (3, 6, 12, 24, …), the sum of terms 1 through 5 is

$S_5$: sum up to term 5 $=$ first term (3) $\times$ 1 minus the ratio to the 5th power ($1 - 2^5 = -31$) $\div$ 1 minus the ratio ($1 - 2 = -1$)

$3 \times (1 - 2^{5}) = 3 \times (-31) = -93$

$-93 \div (1 - 2) = -93 \div (-1) = 93$

Key idea

When the common ratio is greater than 1, both the numerator ($1 - r^n$) and the denominator ($1 - r$) are negative, as in the example. The two negatives cancel in the division, so the answer comes out positive. Textbooks also often use $S_n = \dfrac{a(r^n - 1)}{r - 1}$, which multiplies the top and bottom by $-1$ to flip the signs. It is exactly the same formula, just easier to use when the common ratio is greater than 1. When the common ratio is $1$, the denominator $1 - r$ becomes $0$, so this formula does not work. In that case, use the next formula.

Formula for the sum when the common ratio is $1$

Figure

Standard notation (the usual math form)

$S_n$ $=$ $a$ $\times$ $n$

In words (symbols replaced with words)

③ $S_n$: sum of terms 1 through $n$ $=$ ① $a$: first term $\times$ ② $n$: number of terms

The formula in words

① Multiply the $a$: first term by the

② $n$: number of terms ,

③ and you get the $S_n$: sum of terms 1 through $n$

Quick example

For the geometric sequence with first term 5 and common ratio 1 (5, 5, 5, 5, …), the sum of terms 1 through 4 is

$S_4$: sum up to term 4 $=$ first term (5) $\times$ number of terms (4)

$5 \times 4 = 20$

Key idea

In a geometric sequence with common ratio $1$, every term is the same as the first term (5, 5, 5, …). So the sum is simply "first term × count". Think of it as the special route for $r = 1$, where the second formula does not work (this calculator also switches to this formula automatically when $r = 1$).

Formula for the sum of an infinite geometric series (when the absolute value of the common ratio is less than $1$)

Graph

Standard notation (the usual math form)

$S$ $=$ $a$ $\div$ $(1 - r)$

In words (symbols replaced with words)

③ $S$: sum when you keep adding forever $=$ ① $a$: first term $\div$ ② $(1 - r)$: 1 minus the ratio

The formula in words

① Divide the $a$: first term by

② $(1 - r)$: 1 minus the ratio ,

③ and you get the $S$: sum when you keep adding forever

Quick example

If you keep adding the geometric sequence with first term 100 and common ratio 0.5 (100, 50, 25, 12.5, …) forever, you get

$S$: infinite sum $=$ first term (100) $\div$ 1 minus the ratio (1 − 0.5 = 0.5)

$100 \div (1 - 0.5) = 100 \div 0.5 = 200$

Key idea

This formula works only when the absolute value of the common ratio is less than $1$ ($-1 < r < 1$). Then the terms keep getting closer to $0$, so even if you keep adding forever, the sum gets closer and closer to one fixed value (200 in this example). We say the series converges, and the value it approaches is the sum $S$ of the infinite geometric series. When the absolute value of the common ratio is $1$ or more, the sum grows without limit (or keeps swinging as the signs alternate), so there is no infinite sum. This calculator also shows this value only when the absolute value of the common ratio is less than $1$. Where does the formula come from? In the second formula, as $n$ gets larger and larger, $r^n$ approaches $0$, so $S_n = \dfrac{a(1 - r^n)}{1 - r}$ approaches $\dfrac{a}{1 - r}$.

A geometric sequence keeps multiplying by the same number (the common ratio). The $n$th term is "the first term multiplied by the common ratio $n-1$ times", and the sum is "first term × (1 − common ratio to the $n$th power) ÷ (1 − common ratio)" ("first term × number of terms" when the common ratio is $1$). When the absolute value of the common ratio is less than $1$, the sum when you keep adding forever is $a \div (1 - r)$.

Symbols and terms

Symbols

$a$	a	The first term. The very first term (number) of the sequence. For the sequence 3, 6, 12, 24, …, it is 3.
$r$	ar	The common ratio. The fixed number you multiply by to get the next term. For 3, 6, 12, 24, … it is 2. The letter comes from "ratio", and it can be a decimal or 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 10" is $n = 10$.
$a_n$	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.
$r^{n-1}$	r to the power of n minus one	The common ratio $r$ multiplied by itself $n-1$ times (a power). From term 1 to term $n$ there are $n-1$ steps, so you multiply one time 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".
$S$	S	The sum of the infinite geometric series. The value the sum approaches when you keep adding terms forever. It is also written $S_\infty$ ("S sub infinity").
$\cdots$	dot dot dot (ellipsis)	A symbol that says the pattern continues the same way. Writing 3, 6, 12, 24, … says the numbers keep doubling.

Terms

sequence	A list of numbers in a set order. Each number in the list is called a term.
geometric sequence	A sequence where the ratio between neighboring terms is always the same, such as 3, 6, 12, 24, … (multiply by 2 each time). It is also called a geometric 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 $a$ or $a_1$.
common ratio	In a geometric sequence, the fixed number you multiply by to get the next term. You get the same value whichever neighboring pair you divide (later term ÷ earlier term). It can be a decimal (for example 0.5, halving each time) or a negative number (the signs alternate).
power	A number multiplied by itself several times. $2^5$ (2 to the 5th power) is $2 \times 2 \times 2 \times 2 \times 2 = 32$. The small number at the upper right (the exponent) tells how many times to multiply. Taught in Grade 6.
general term	The $n$th term written as a formula in $n$, also called the explicit formula. For a geometric sequence it is $a_n = a r^{n-1}$. Put a position number in for $n$ and you get that term right away.
infinite geometric series	The terms of a geometric sequence added up forever. Only when the absolute value of the common ratio is less than $1$ does the sum converge to a fixed value, $\dfrac{a}{1-r}$. It is taught in Algebra 2 or precalculus, but the idea is simple: each amount you add gets smaller and smaller.
converge	To get closer and closer to one fixed value the more you add (or the further you go along the sequence). 100 + 50 + 25 + 12.5 + … gets closer to 200 the more you add, so we say it converges to 200.
arithmetic sequence	A sequence where the difference between neighboring terms is always the same, such as 5, 8, 11, 14, … (add 3 each time). It grows differently from a geometric sequence, which multiplies by 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 doubles each time" in a list of numbers like 3, 6, 12, 24, … Being able to make a table that pairs each position number with its value
Multiplying decimals and fractions (Grades 5–6)	Knowing that multiplying by 0.5 halves a number, as in $64 \times 0.5 = 32$ Knowing that multiplying by a number less than 1 gives an answer smaller than what you started with
Negative numbers (Grade 7)	Being able to multiply two negative numbers, as in $(-2) \times (-2) = 4$ (this is why the signs alternate when the common ratio is negative) Being able to divide two negative numbers, as in $(-93) \div (-1) = 93$
Exponents and variables (Grade 6)	Being able to work out powers such as $2^5 = 32$ (multiply as many times as the small number at the upper right says) Being able to substitute, such as putting $n = 10$ into $n - 1$ to get 9
Order of operations (Grades 5–6)	Doing powers and what is inside parentheses first (in $3 \times (1 - 2^5)$, do $2^5$, then the parentheses, then the multiplication)

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 3, common ratio 2, 10 terms)

First term a	3
Common ratio r	2
Number of terms n	10
nth term	=B1*B2^(B3-1)

Table to find the sum of terms 1 through n

First term a	3
Common ratio r	2
Number of terms n	10
Sum of terms 1 through n	=IF(B2=1,B1B3,B1(1-B2^B3)/(1-B2))

Table to find the sum of an infinite geometric series (when the absolute value of the common ratio is less than 1)

First term a	100
Common ratio r	0.5
Sum when you keep adding forever	=B1/(1-B2)

After pasting, the upper cells are the inputs and the bottom cell shows the calculated result.
In the formulas, "B1" and "B2" say "use the number in that cell", "*" is multiplication, "/" is division and "^" is a power (B2^(B3-1) is "the common ratio to the power of (number of terms − 1)").
In the second table, IF(B2=1, …) splits the cases: "if the common ratio is 1, use first term × number of terms; otherwise use the sum formula" (with a common ratio of 1, the denominator of the sum formula would be 0).
For example, the first table shows 1536 (term 10) in B4, the second table shows 3069 (the sum) in B4, and the third table shows 200 in B3. Just replace the input cells 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 3, common ratio 2, 10 terms)

First term a	3
Common ratio r	2
Number of terms n	10
nth term	=B1*B2^(B3-1)

Table to find the sum of terms 1 through n

First term a	3
Common ratio r	2
Number of terms n	10
Sum of terms 1 through n	=IF(B2=1,B1B3,B1(1-B2^B3)/(1-B2))

Table to find the sum of an infinite geometric series (when the absolute value of the common ratio is less than 1)

First term a	100
Common ratio r	0.5
Sum when you keep adding forever	=B1/(1-B2)

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

How to calculate it in Python

first_term = 3        # first term
common_ratio = 2      # common ratio
number_of_terms = 10  # number of terms

# nth term = first term * common ratio to the power (n - 1)
nth_term = first_term * common_ratio ** (number_of_terms - 1)
# sum of terms 1 through n = first term * (1 - ratio^n) / (1 - ratio); when the ratio is 1, first term * n
if common_ratio == 1:
    sum_of_terms = first_term * number_of_terms
else:
    sum_of_terms = first_term * (1 - common_ratio ** number_of_terms) / (1 - common_ratio)
# first 10 terms of the sequence (for checking)
sequence = [first_term * common_ratio ** k 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 ratio, number of terms) and run it. This example prints 1536 for term 10 and 3069.0 for the sum. To also get the sum of the infinite geometric series when the absolute value of the common ratio is less than 1, add the line "print(first_term / (1 - common_ratio))" at the end.

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

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

aₙ = a × rⁿ⁻¹

a_n = a r^{n-1}

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

a_n = a r^(n-1)

a*r^(n - 1)

an := a*r^(n - 1);

a_n = a*r^(n - 1);

a_n = ar^(n - 1)

Formula for the sum of the first $n$ terms (when the common ratio is not $1$)

Sₙ = a(1 − rⁿ) ÷ (1 − r)

S_n = \dfrac{a(1 - r^{n})}{1 - r} \quad (r \neq 1)

<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>a</mi>
        <mo>(</mo>
        <mn>1</mn>
        <mo>&#x2212;</mo>
        <msup><mi>r</mi><mi>n</mi></msup>
        <mo>)</mo>
      </mrow>
      <mrow>
        <mn>1</mn>
        <mo>&#x2212;</mo>
        <mi>r</mi>
      </mrow>
    </mfrac>
  </mrow>
</math>

S_n = (a(1 - r^n))/(1 - r)

a*(1 - r^n)/(1 - r)

Sn := a*(1 - r^n)/(1 - r);

S_n = a*(1 - r^n)/(1 - r);

S_n = a(1 - r^n)/(1 - r)

Formula for the sum when the common ratio is $1$

Sₙ = a × n

S_n = a n \quad (r = 1)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>S</mi><mi>n</mi></msub>
    <mo>=</mo>
    <mi>a</mi>
    <mi>n</mi>
  </mrow>
</math>

S_n = a n

a*n

Sn := a*n;

S_n = a*n;

S_n = an

Formula for the sum of an infinite geometric series (when the absolute value of the common ratio is less than $1$)

S = a ÷ (1 − r)

S = \dfrac{a}{1 - r} \quad (|r| < 1)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>S</mi>
    <mo>=</mo>
    <mfrac>
      <mi>a</mi>
      <mrow>
        <mn>1</mn>
        <mo>&#x2212;</mo>
        <mi>r</mi>
      </mrow>
    </mfrac>
  </mrow>
</math>

S = a/(1 - r)

a/(1 - r)

S := a/(1 - r);

S = a/(1 - r);

S = a/(1 - r)

How to have ChatGPT do the calculation

You are an assistant for geometric 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 × r^(n-1), the sum with S_n = a(1 - r^n)/(1 - r) (S_n = a × n when r = 1), and the sum of the infinite geometric series with S = a/(1 - r) when |r| < 1.

1. Term 10 and the sum of terms 1 through 10 of the geometric sequence with first term 3 and common ratio 2
2. Term 6, the sum of terms 1 through 6, and the sum when you keep adding forever, for the geometric sequence with first term 100 and common ratio 0.5
3. Term 6 and the sum of terms 1 through 6 of the geometric sequence with first term 3 and common ratio -2

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

How to Use

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Enter your numbers

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

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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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\(a\)	a	The first term. The very first term (number) of the sequence. For the sequence 3, 6, 12, 24, …, it is 3.
\(r\)	ar	The common ratio. The fixed number you multiply by to get the next term. For 3, 6, 12, 24, … it is 2. The letter comes from "ratio", and it can be a decimal or 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 10" is \(n = 10\).
\(a_n\)	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.
\(r^{n-1}\)	r to the power of n minus one	The common ratio \(r\) multiplied by itself \(n-1\) times (a power). From term 1 to term \(n\) there are \(n-1\) steps, so you multiply one time 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".
\(S\)	S	The sum of the infinite geometric series. The value the sum approaches when you keep adding terms forever. It is also written \(S_\infty\) ("S sub infinity").
\(\cdots\)	dot dot dot (ellipsis)	A symbol that says the pattern continues the same way. Writing 3, 6, 12, 24, … says the numbers keep doubling.

Geometric 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

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