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# Geometric sequence calculator and problems solver

This online calculator solves common geometric sequence problems.

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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