# Geometric sequence calculator and problems solver

This online calculator solves common geometric sequence problems.

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

1. Find the n-th term of a geometric sequence given the m-th term and the common ratio. Example problem: A geometric sequence with a common ratio equals -1, and its 1-st term equals 10. Find its 8-th term.

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

The detailed description of the solutions is shown through geometric sequence theory underneath the calculator, as always.

#### 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 the 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 a common ratio using the following formula derived from the division of equation for one known term by an 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 convenience, the calculator above also calculates the first term and general formula for the n-th term of a geometric sequence.

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PLANETCALC, Geometric sequence calculator and problems solver