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Common Ratio

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22nd September 2020

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Introduction

A type of sequence is a geometric sequence. 

It is a group of numbers that is ordered with a specific pattern ball bouncing is an example of a finite geometric sequence.

Each time the ball bounces its height gets cut down by half. If the ball's first height is 4 feet, the next time it bounces it's the highest bounce will be at 2 feet, then 1, then 6 inches, and so on, until the ball stops bouncing.

Common Ratio in real life

The last row has 56 seats

A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value.

Also read:

The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (that is, if you find the ratio of) successive terms, you'll always get this value.


How to Find Common Ratio

The geometric sequence is based upon the multiplication of a constant value to arrive at the next term in the sequence.

Geometric sequences follow a pattern of multiplying a fixed value (non zero) from one term to the next.

The number has to be multiplied each time it is constant (always the same).

\(\begin{align}{a_1},({a_1}r),({a_1}{r^2}),({a_1}{r^3}),({a_1}{r^4}),....\end{align}\)

The fixed value is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term gives in the common multiple.

So, divide the second term by the first term.

Geometric sequence

Geometric sequences have a non-linear nature when plotted on the graph (as a scatter plot).

The graph x-axis consists of the counting numbers 1, 2, 3,  ... (representing the location of each term), and the range containing the actual terms of the sequence.

In the graph shown above, while the x-axis increased by a constant value of one, the y value increased by a constant multiple of 3.


Examples: How to Find Common Ratio

Take a scenario where you could have a starting number of 4 and a common ratio of 3.

Note that the first term will be the starting number.

So the first term is:4

Now the second term is going to be equal to the first term (4) multiplied by the common ratio (3); this equals 12.

So the second term is 12, and the first two terms are: 4, 12

Again the third term is going to be equal to the second term (12) multiplied by the common ratio (3); this equals 36.

So the third term is 36, and the first three terms are: 4, 12, 36

The same for the fourth term is going to be equal to the third term (36) multiplied by the common ratio (3); this equals 108.

So the fourth term is 108, and the first four terms are: 4, 12, 36, 108

Did you notice how geometric sequences grow very fast!

Geometric Sequence: Common Ratio, r:
3, 12, 48, 192, 768,...   r = 4. A 4 is multiplied times each term to arrive at the next term.so ... divide a2 by a1 to find the common ratio of 4.
-6, 18, -54, 162, -486, ...  r = -3. A -3 is multiplied times each term to arrive at the next term.so... divide a2 by a1 to find the common ratio of -3.
Number sequence ExpressionA 2/3 is multiplied times each term to arrive at the next term.so... divide a2 by a1 to find the common ratio of 2/3.

The second Scenario assumes that your friend has the flu virus, and he forgot to cover his mouth when two friends came to visit while he was sick in bed.

They leave, and the next day they also have the flu. Let's assume that each friend in turn spreads the virus to two of their friends by the same droplet spread the following day.

Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner:

Virus Tree: How to find Common Ratio?

Each one infects two more people with the flu virus.

The above diagram represents the number of infected people after n days since your friend first infected your 2 friends.

Now the virus is carried over to 2 people who start the chain (a=2). The next day, each one then infects 2 of their friends. Now 4 people are newly-infected.

Each of them infects 2 people on the third day, and 8 new people are infected, and so on.

These  process of events can be written as a geometric sequence:

2; 4; 8; 16; 32…

Note the constant ratio (r=2) between the events. Recall from the linear arithmetic sequence how the common difference between terms was established.

In the geometric sequence, we can determine the constant ratio (r) from T2/T1=T3/T2=r

More generally we find the formula for common ratio = Tn/Tn-1.

Let's see a few more examples…..


Common Ratio Problems: Finding Common Ratio

Example 1

 

 

What is  the common ratio for the geometric sequence: 

1,2,4,8,16,...

Solution:

Divide each term by the previous term to determine whether a common ratio exists.

\(\begin{align}\frac{2}{1} = 2\frac{4}{2} = 2\frac{8}{4} = 2\frac{{16}}{8} = 2\end{align}\)

The sequence is geometric because there is a common multiple.

The answer is 2.
Example 2

 

 

What is  the common ratio for the geometric sequence: \(\begin{align}3, - 1,\frac{1}{3}, - \frac{1}{9},...\end{align}\)

Solution:

The ratio, r, can be found by dividing the second term by the first term, which in this example yields -1/3.

It clearly shows that multiplying each term by -1/3 will create the next term.
Example 3

 

 

What is the common ratio for a geometric sequence whose formula is: an = 4(3)n-1.

Solution:

A listing of the terms will show what is happening in the sequence (start with n = 1). 

4, 12, 36,108,...

The common difference is 3. The position of the 3 in the formula also indicates its importance as the common ratio.

Summary

We understand from the above examples that one of the two simplest sequences to work with is a geometric sequence.

It is used throughout mathematics. It has important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

A key part of the geometric sequence is its common ratio.

The above blog showcased using examples of how to find common ratio in various scenarios.  

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FAQs

What is common ratio?

A sequence is a set of numbers. It can be a group that is in a particular order, or it can be just a random set. A geometric sequence is a group of numbers that is ordered with a specific pattern. The pattern is determined by a certain number that is multiplied to each number in the sequence. This determines the next number in the sequence. The number multiplied must be the same for each term in the sequence and is called a common ratio.

What is the common ratio of the sequence? -2, 6, -18, 54,...?

6/-2 = 3

What is the geometric ratio?

In mathematics, a geometric progression, also known as a geometric sequence, 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. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with ratio 3.


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