Geometry Series Formula

Understanding Geometry Series

Definition of sequences Geometry is a sequence which each tribe is obtained from the result of multiplying the previous term with a certain constant. A geometry series is a sequence that satisfies the properties of a result for a term with a previous sequence of values of constants.

For example, the geometry sequence is a, b, and c, then c / b = b / a is equal to a constant. The results for the adjacent tribe are called ratio (r).

For example a geometry series is found like the following:

U1, U2, U3, …, Un-1, Un
Then U2 / U1, U3 / U2, …, Un / Un-1 = r (constant or ratio)
Then how to determine the nth term of the geometry sequence? See the following explanation:

U3 / U2 = r then U3 = U2.r = a.r.r = ar2
Un / Un-1 = r then Un = Un-1. r = arn-2.r = arn-2 + 1 = arn-1
so it can be concluded that the geometry of the geometry n rows is Un = arn-1

a = initial rate r ratio.

Geometry Series Formula

the sum of the first n terms of a geometry sequence is called a geometric sequence. If the nth term of the geometry sequence is formulated: an = a1rn – 1, then the geometry series can be written as,

If we multiply the series with -r then add it to the original series, we get

So we get Sn – rSn = a1 – a1rn. By solving this equation for Sn, we get

The result above is the formula for the number of first n terms of the infinite geometry sequence.

Number n First Tribe Geometry
Given a geometric sequence with the first term a1 and the ratio r, the number of n the first term is

Or it can be said: The sum of the geometric sequences equals the difference from the first term and the term n + 1, then divided by 1 minus the ratio.

Example of Geometry Series Problems

Problem: Calculate the number of the first 9 terms of the sequence an = 3n .