Find the 6th term of the geometric sequence calculator

How does this geometric sequence calculator work?

In mathematics, a geometric progression is also known as geometric sequence and represents a sequence of numbers (sequence being an ordered list of numbers) with the particularity that each member/term excepting the first one is found by multiplying the previous one by a fixed, non-zero number generally called the common ratio.

For example, the sequence 2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250 is a geometric progression with the common ratio being 5.

The formulas applied by this geometric sequence calculator are detailed below while the following conventions are assumed:

- the first number of the geometric progression is a;

- the step/common ratio is r;

- the nth term to be found in the sequence is an;

- The sum of the geometric progression is S.

Then:

an = arn-1

If r ≠ 1 then S = [a(1-rn]/(1-r)

If r = 1 then S = an

Example of a geometric progression calculation

Let’s take an example of a geometric progression having first number a= 2, r = 3 for which we try to figure out which is the 10th number in the sequence:

■ The 10th value of the sequence (a10) is 39,366

■ Sample of the first ten numbers in the geometric sequence: 2; 6; 18; 54; 162; 486; 1,458; 4,374; 13,122; 39,366

■ Sum of all numbers until the 10th: 59,048

11 Jun, 2015

About Geometric Sequence Calculator

This Geometric Sequence Calculator is used to calculate the nth term and the sum of the first n terms of a geometric sequence.

FAQ

In mathematics, a geometric sequence, also known as a 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. The sum of the numbers in a geometric progression is also known as a geometric series.

If the initial term of a geometric sequence is a1 and the common ratio is r, then the nth term of the sequence is given by:

an = a1rn-1

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

Problem type

Find term by another term and common ratio

Find term by two another terms

First Term of the Geometric Sequence

nth Term of the Sequence Formula

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

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

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

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.

How do you find the sixth term of a geometric sequence?

What is the sum of the geometric sequence 6 if there are 6 terms?