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 .
The number of the first 9 terms can also be denoted in the sigma notation as follows.
From the series we can get the first term a1 = 3 , the ratio r = 3, and the number of terms n = 9. Using the formula for the number n first term, we get
So, the number of the first nine terms of the sequence an = 3n is 29,523.
Well, it’s easy not to calculate geometric series and infinite geometry sequences above? We feel that the discussion of geometric series formulas along with examples of geometry sequences and the answers to their discussion can be written this time. Hopefully what we have learned in this article can be useful especially for those of you who are studying math material.