**Math Calculators** ▶ Geometric Sequence Calculator

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An online geometric sequence calculator helps you to find geometric Sequence, first term, common ratio, and the number of terms. This geometric series calculator provides step-wise calculation and graphs for a better understanding of the geometric series. Continue reading to learn how to find common ratios, the sum of finite geometric series, and much more.

In mathematics, a geometric sequence is also called a geometric progression. It is defined as a list of numbers in which each item in the Sequence is multiplied by a non-zero constant called the general ratio “r”.

In other words, a geometric sequence or progression is an item in which another item varies each term by a common ratio. When we multiply a constant (not zero) by the previous item, the next item in the Sequence appears. It is expressed as follows:

$$ a, a(r), a(r)^2, a(r)^3, a(r)^4, a(r)^5and so on. $$

where a is the first item and r is a common ratio.

Note: It should be noted that if we separate the next item from the previous item, then we will get a value corresponding to the common ratio.

Suppose we divide the third term by second term, and we get:

$$ A(r)^3/a(r)^2 = r $$

The general form of Geometric Progression is:

$$ a, a(r), a(r)^2, a(r)^3, a(r)^4, …, a(r)^n $$

Where,

an = nth term

a = First term

r = common ratio

However, an online Arithmetic Sequence Calculator allows you to compute the nth value, sum, and Arithmetic sequence.

**Example: **

**Find Geometric Sequence, when the first term is 2, common ratio “r” is 4, and the number of terms is 10.**

**Solution: **

A_1 = 2, r = 4, n = 10

The geometric sequence calculator finds the Nth term of geometric Sequence is:

A_n = a_1 * r{n-1}

A_{10} = 2 * (4)^{10-1}

A_{10} = 2 * 4^9

A_{10} =2 * 262144

A_{10} = 524288

**The sum of geometric series calculator find the sum of the first n-terms:**

S_n = a_1 * (1 – r^n) / 1 – r

S_{10} = 2 * (1 – 4^{10}) / 1 – 4

S_{10} = 2 * (1 – 1048576) / -3

S_{10} = 699050

The sum of the common ratio calculator determines the first ten terms of the Sequence are:

2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, ….

Suppose “r” be the common ratio and “a” be the first item for a Geometric progression.

Then the second item of the series, \( a^2 = a * r = ar \)

Third item, \( a^3 = a^2 * r = a(r) * r = a(r)^2 \)

Therefore, nth term, \( a_n = ar^{n-1} \)

Hence, the geometric sequence formula used by the geometric sequence calculator to find the nth term of Geometric series is:

$$ A_n = ar^{n-1} $$

First term is = a Consider the series is \( a, ar, a(r)^2, a(r)^3, a(r)^4…… \)

Second term is = ar

Similarly, the nth item is, \( k_n = ar^{n-1} \)

Thus,

Common ratio = (Any item) / (Preceding item)

$$ = k_n / k_{n-1} $$

$$ = (ar^{n – 1} ) /(ar^{n – 2}) = r $$

Thus, the general term of a Geometric progression is given by \( ar^{n-1} \) and the general form of a Geometric sequence is \( a + a(r) + a(r)^2 + a(r)^3 + ….. \)

**Example: **

**Find the common ratio, where first term a_1 = 2 and a_3 =16. **

**Solution: **

The common ratio calculator uses a simple formula for determining the ratio:

$$ R = ^{n-1} \sqrt { a_n / a_1} $$

$$ R = ^{3-1} \sqrt { 16 / 2} $$

$$ R = ^{2} \sqrt { 8} $$

$$ R = 2.82842712 $$

Let’s \( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \) is the given Geometric series.

Then the sum of finite geometric series is:

$$ S_n = a + ar + a(r)^2 + a(r)^3 + a(r)^4 + …+ar^{n-1} $$

The formula to determine the sum of n terms of Geometric sequence is:

$$ S_n = a[(r^n-1)/(r-1)] if r ≠ 1 $$

Where a is the first item, n is the number of terms, and r is the common ratio.

Also, if the common ratio is 1, then the sum of the Geometric progression is given by:

S_n = na if r=1

However, an online Geometric Mean Calculator allows you to compute the geometric mean for a given data set of percentages or numbers.

**Example:**

**Find the sum of first n-terms of geometric sequence, where first term (a_1) = 2, common ration (r) = 2, and sum of first n-terms = 4. **

**Solution:**

The geometric sequence formula calculator finds the sum of geometric series by:

$$ S_n = a_1 . (1 – r^n) / 1 – r $$

The sum of geometric series sum calculator substitutes the given values in formula:

$$ S_n = a_1 . (1 – r^n) / 1 – r $$

$$ 4 = 2 * (1 – 2^n) / 1 – 2 $$

$$ (1 – 2^n) = 4 / 2 * (1 – 2) $$

$$ (1 – 2^n) = – 2 $$

$$ 2^n = 3 $$

$$ Log (2^n) = log (3) $$

$$ n . log (2) = log (3) $$

$$ n = 1.5849 $$

The terms of a finite Geometric progression can be written as \( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \)

And the items \( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \), is called finite geometric series.

The sum of finite Geometric series is given by:

$$ S_n = a[(r^n-1)/(r-1)] if r ≠ 1 $$

Terms of an infinite Geometric Progression can be written as \( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1},……. \)

\( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1},……. \) is called infinite geometric series.

**Example: **

**Find the sum of the infinite geometric series 64 + 32 + 16 + 8 + 4 + 2 ****⋯ **

**Solution:**

First, the infinite geometric series calculator finds the constant ratio between each item and the one that precedes it:

$$ R = 32/64 $$

$$ =1 / 2 $$

Now, geometric sequence calculator substitute r=1/2 and a=64 into the formula for the sum of an infinite geometric series:

$$ s=64 / (1−1/2) = 64 / (1/2) = 128 $$

Below is a list of geometric progression formulas that can help to solve the various types of problems.

- The general forms of GP terms are a, ar, a(r)^2, a(r)^3, a(r)^4, etc., where a is the first term and r is the common ratio.
- The nth term of Geometric sequence is k_n = ar^{n-1}
- Common ratio = r = k_n/ k^{n-1}
- The geometric sequence formula to determine the sum of the first n terms of a Geometric progression is given by:

S_n = a[(r^n-1)/(r-1)] if r > 1 and r ≠ 1

S_n = a[(1 – r^n)/(1 – r)] if r < 1 and r ≠ 1

- The nth item at the end of GP, the last item is l, and the common ratio is r = l / [r (n – 1)].
- The sum of infinite series, that is the sum of Geometric Sequence with infinite terms is S∞ = a / (1-r) such that 1 >r >0.
- If there are 3 values in Geometric Progression, then the middle one is known as the geometric mean of the other two items.
- If a, b, and c are three values in the Geometric Sequence, then “b” is the geometric mean of “c” and “a”. This can be written as b = √ac or b^2 = ac
- Assume that “r” and “a” are the common ratio and first term of a finite geometric sequence with n terms. Therefore, the kth item at the end of the geometric series will be ar^{n – k}.

An online geometric calculator determines different geometric terms by following these steps:

- First, choose an option from the drop-down list in order to find any term of geometric Sequence.
- Now, substitute the corresponding values according to your selection.
- Click on the calculate button to see the results.

The geometric sum calculator provides the step-by-step solution and calculates:

- Geometric Sequence: find the n-th terms, the sum of n terms, sequence of n terms, and display a graph.
- First Term of the Sequence
- Common Ratio
- Nth term and the sum of geometric series

Types of Series and Sequence

- Arithmetic Sequences.
- Harmonic Sequences.
- Geometric Sequences.
- Fibonacci Numbers.

If the continuous terms are in a constant ratio, then the Sequence is geometric. On the other hand, if there is a constant difference between consecutive elements, then the Sequence is called an arithmetic sequence.

Use this online geometric sequence calculator to evaluate the nth term and the sum of the first n terms of the geometric sequence. It displays complete calculations for finding the sequence, sum of series, and common ratios. This calculator provides all calculations in a fraction of a second using the geometric sequence formula.

From the source of Wikipedia: Geometric progression, Elementary properties, Derivation, Complex numbers, Product.

From the source of Lumen Learning: Definition of geometric progression, Behavior, Summing the First n Terms in a Geometric Sequence, Infinite Geometric Series, Applications of Geometric Series, Repeating Decimal, Archimedes’ Quadrature of the Parabola, Fractal Geometry.

From the source of Purple Math: Find the common difference, Find the common ratio, arithmetic, and geometric sequences, adding (or subtracting) the same values.