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Geometric Sequence Calculator

Geometric Sequence Calculator

Find:

First Term (a₁)

Common Ratio (r)

Number of Terms (n)

Enter n-th term a(n)

Sum of first n-terms S(n)

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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.

What is the Geometric Sequence?

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:

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

Geometric Progression or Geometric Sequence:

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, ….

Nth Term of Geometric Sequence:

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:

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

How to Find 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:

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

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

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

$$ R = 2.82842712 $$

How to Find the Sum of a Geometric Series?

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:

$$ S_n = a_1 . (1 – r^n) / 1 – r $$

$$ 4 = 2 * (1 – 2^n) / 1 – 2 $$

$$ (1 – 2^n) = 4 / 2 * (1 – 2) $$

$$ (1 – 2^n) = – 2 $$

$$ 2^n = 3 $$

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

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

$$ n = 1.5849 $$

Infinite Geometric Sequence:

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:

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

Geometric Progression Formulas:

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}.

How Geometric Sequence Calculator Works?

An online 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:

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

FAQ:

What are the main types of Sequence?

Types of Series and Sequence

  • Arithmetic Sequences.
  • Harmonic Sequences.
  • Geometric Sequences.
  • Fibonacci Numbers.

How ​​to know if it is arithmetic or geometric sequence?

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.

Conclusion:

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.

Reference: 

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.