Math Calculators ▶ Geometric Sequence Calculator
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Table of Content
Our 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.
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.
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
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:
$$ S_n = a_1 . (1 – r^n) / 1 – r $$
$$ 4 = 2 * (1 – 2^n) / 1 – 2 $$
$$ (1 – 2^n) = 4 / 2 * (1 – 2) $$
$$ (1 – 2^n) = – 2 $$
$$ 2^n = 3 $$
$$ Log (2^n) = log (3) $$
$$ n . log (2) = log (3) $$
$$ n = 1.5849 $$
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
An online geometric calculator determines different geometric terms by following these steps:
The geometric sum calculator provides the step-by-step solution and calculates:
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.
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