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**Table of Content**

Calculate the geometric sequence, first term \(a_{1}\), common ratio (r), number of terms, and total sum with the geometric sequence calculator.

The geometric sequence solver provides step-wise calculations and graphs for a better understanding of the geometric series.

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'”**

When we multiply a constant (not zero) by the previous item, the next item in the Sequence appears.

\(a, a(r), a(r)^2, a(r)^3, a(r)^4, a(r)^5, …\)

where:

**a** = first item

**r** = common ratio.

Thus, the general term of a Geometric progression is given by \( ar^{n-1} \) and the general form of a Geometric sequence is \( a + a(r) + a(r)^2 + a(r)^3 + ….. \). The sum of geometric sequence calculator finds the nᵗʰ term and the sum of a geometric sequence from the first term to infinity.

Let \(a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \) 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:

\(S_{n} = na if r=1\)

The common ratio finder calculates the common ratio in the geometric sequence to determine all the terms.

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 the 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 sequences 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}.

The geometric sequence formula calculator computes the common ratio, the first and the nth terms of the geometric sequence.

Find the common ratio, where the first term \(a_1\) = 2 and \(a_3\) =16.

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

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

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

\(R = ^{2} \sqrt { 8}\)

\(R = 2.82842712\)

The geometric calculator determines different geometric terms by following these steps:

**Input:**

- 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.

**Output:**

- Geometric Sequence
- 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

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

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