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# Geometric sequence calculator and problems solver

This online calculator solves common geometric sequences problems.

### This page exists due to the efforts of the following people:

#### Timur

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.

#### 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.

PLANETCALC, Geometric sequence calculator and problems solver