With so many different elements in mathematics, it becomes difficult to ace every single one. Whilst you are focusing on Pythagoras’s theorem, geometric progression gets left on the back burner. Don’t worry, we are here to help you understand what geometric progressions are and how to calculate them. In this guide, we explain everything you need to know, from how to calculate the next number in a geometric sequence, to the ins and outs of what a common ratio actually means.
What is a geometric progression?
A geometric progression is a special kind of progression/sequence. For a sequence of numbers to be considered a geometric progression, they must all be multiplied by a common ratio. The previous number in the sequence multiplied by the common ratio will provide you with the next number in the geometric series.
For example, if your common number was 2, the sequence would be as follows:
2, 4, 8, 16, 32…
We can observe the geometric progression as each successive term is the previous term multiplied by two. So, 2 x 2 = 4 then 4 x 2 = 8 then 8 x 2 = 16 then 16 x 2 = 32.
Why is it called a geometric progression?
If you thought geometric progression had something to do with shapes and geometry, you wouldn’t be the first. So why exactly is geometric progression called geometric progression? Well, the answer is partially related to geometry, even if it might not seem so at first.
The term geometric progression comes from the idea of a geometric mean. The geometric mean helps to create perfect right-angle triangles, as each previous side length is multiplied by the common ratio to increase the length of a triangle’s side at regular intervals. Therefore, although we are primarily dealing with numbers when solving the sequence, they can be used for geometrical purposes as well, hence the name.
What is the difference between geometric progression and arithmetic progression?
There are many differences between a geometric sequence and an arithmetic sequence, although people often get confused between the two. Below is a breakdown of the key differences:
- GP has a common ratio, AP doesn’t
- AP has a common difference, GP doesn’t
- The next in the sequence is the product of the previous number and common ratio
- The next in the sequence is the sum of the previous number and the common difference
- The GP variation is non-linear
- The AP variation is linear
How many geometric progressions are possible?
An infinite number of geometric progressions are possible within any given sequence. Although the initial term might be anything as small as 2, continually multiplying by the common ratio will allow you to increase the number indefinitely. There are therefore infinite terms possible for progression sequences. However, there are some types of GP that are specifically defined as finite progressions.
What are the properties of geometric progression (GP)?
There are some common properties of all geometric progressions, which help us to group the sequences together and better understand them. Below are the key properties you need to know:
- If you multiply a non-zero quantity by each term of the GP, then the resulting sequence shares the common difference and will also be a GP
- If you divide a non-zero quantity by each term of the GP, then the resulting sequence shares the common difference and will also be a GP
- Reciprocals of all the terms in GP also form a GP
- If all the terms in a GP are raised to the same power, then the new series is also in GP
- If y² = xz, then the three non-zero terms x, y and z are in GP
Geometric progression formulas
nth term
The n-th term formula looks like this:
an = arn – 1 (or) an = r an – 1
All you need to find the nth term of a sequence is the initial term and the common ratio. Within the formula, each letter stands for the following:
a = the initial term
r = the common ration
n = the consecutive terms that you want to calculate
This is one of the most useful formulas when calculating terms in a GP. This is because it does not need to be used to calculate adjacent terms, but instead can be used to calculate any term within the sequence. You can use it to find the 3rd term, the 8th term the 15th term, or even the 100th term, so you can go far beyond just neighboring terms.
Why is it important to follow the order of operations?
Geometric progression sum formula
If you need to calculate the sum of all the numbers in your sequence, you can use the geometric progression sum formula. The sum formula is different depending on whether you are calculating the sum of a finite or infinite series.
The finite sum formula is as follows:
Sn = a(1 − rn)/(1 − r) for r ≠ 1, and
Sn = an for r = 1
The infinite sum formula is as follows:
S∞ = a/(1 – r), when |r| < 1
The sum cannot be found when |r| ≥ 1
Math tutoring services
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