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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.
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 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:
Output:
Types of Series and Sequence
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.