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Geometric sequence calculator and problems solver

This online calculator solves common geometric sequences problems.

Creative Commons Attribution/Share-Alike License 3.0 (Unported)

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/8312/. Also, please do not modify any references to the original work (if any) contained in this content.

This online calculator can solve geometric sequences problems. Currently, it can help you with the two common types of problems:

  1. Find the n-th term of an geometric sequence given m-th term and the common ratio. Example problem: An geometric sequence has a common ratio equals to -1 and its 1-st term equals to 10. Find its 8-th term.

  2. Find the n-th term of an geometric sequence given i-th term and j-th term. Example problem: An geometric sequence has its 3-rd term equals to 1/2 and its 5-th term equals to 8. Find its 8-th term.

Some theory and description of the solutions can be found below the calculator.

PLANETCALC, Geometric sequence calculator and problems solver

Geometric sequence calculator and problems solver

First Term of the Geometric Sequence
 
Common Ratio
 
nth Term of the Sequence Formula
 
Unknown Term equals to
 

Geometric sequence

To recall, an geometric sequence, or geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Thus, the formula for the n-th term is

a_n=a_1r^{n-1}

where r is the common ratio.

You can solve first type of problems listed above by calculating the first term a1, using the formula

a_1=\frac{a_n}{r^{n-1}}

and then using the geometric sequence formula for the unknown term.

For the second type of problems, first you need to find common ratio using the following formula derived from the division of equation for one known term by equation for another known term

\frac{a_n}{a_m}=\frac{a_1r^{n-1}}{a_1r^{m-1}} \implies \frac{a_n}{a_m}=\frac{r^{n-1}}{r^{m-1}} \implies \frac{a_n}{a_m}=r^{n-m} \implies r=\sqrt[n-m]{\frac{a_n}{a_m}}

After that, it becomes the first type of problem.

For the convinience, the calculator above also calculates the first term and general formula for the n-th term of an geometric sequence.

Creative Commons Attribution/Share-Alike License 3.0 (Unported) PLANETCALC, Geometric sequence calculator and problems solver

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