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# Common Ratio|Definition & Meaning

## Definition

In a **geometric sequence,** the **ratio between** each pair of **consecutive** sequence **elements** remains constant. This ratio is called the **common ratio.**

There are **three types** of fundamental **sequences** in mathematics called **arithmetic, geometric** and **harmonic** sequences. The **geometric sequences** are **multiplicative** sequences.

Such types of sequences have the **property** that each of the terms is a **scalar multiple** of the **previous term.** This scalar multiple is called the **common ratio. **

A **geometric sequence** can be completely defined by any of the **elements** and the **common ratio.** Theoretically, the whole sequence is **reconstructed** using these **two numerical values** only.

## Mathematical Form of the Common Ratio

Let us say that we have a **sequence** of **numbers** as shown below:

{ 1, 3, 9, 27, …… }

The following figure displays this sequence:

**Figure 1: Common Ratio in Geometric Sequences**

**Let us calculate** the **ratio** of the **2nd** and **1st elements:**

\[ \dfrac{\text{2nd element}}{\text{1st element}} \ = \ \dfrac{ 3 }{ 1 } \ = \ 3 \]

Now let us **calculate** the **ratio** of the **3rd** and **2nd elements:**

\[ \dfrac{\text{3rd element}}{\text{2nd element}} \ = \ \dfrac{ 9 }{ 3 } \ = \ 3 \]

You may notice that the** ratio of the 2nd and 1st** elements is **equal** to the **ratio of the 3rd and 2nd** elements. We can see that the **ratio of each consecutive pair** of elements in the above sequence equals 3. Since the **ratio is constant**, we can say that the given sequence is a **geometric sequence,** and the ratio that stays constant is called its **common ratio.**

The above sequence can be **completely described** by the **1st element** (1) and the common ratio (3). Given these two values, we can **construct the sequence** as follows:

{ 1, 1(3), 1(3)(3), 1(3)(3)(3), 1(3)(3)(3)(3), ……. }

It can be **reduced** to:

\[ \{ \ 1(3)^0, \ 1(3)^1, \ 1(3)^2, \ 1(3)^3, \ 1(3)^4, \ ……. \ \} \]

Which is equal to the **original sequence.**

The above version of the given sequence leads to the **general form** of a **geometric sequence** which is given below:

\[ a_n \ = \ a_1 \ r^{ n – 1 } \]

Here, $ a_n $ is the $ n^{th} $ element, $ a_1 $ is the $ 1^{st} $ element and $ r $ is the **common ratio.** Using this formula, we can write the **expression** for the $ (n-1)^{th} $ element:

\[ a_{ n – 1 } \ = \ a_1 \ r^{ n – 2 } \]

**Dividing** above **two equations,** we can derive the **expression for the common ratio**:

\[ \dfrac{ a_n }{ a_{ n – 1 } } \ = \ \dfrac{ a_1 \ r^{ n – 1 } }{ a_1 \ r^{ n – 2 } } \]

\[ \dfrac{ a_n }{ a_{ n – 1 } } \ = \ r^{ n – 1 – ( n – 2 ) } \]

\[ \dfrac{ a_n }{ a_{ n – 1 } } \ = \ r \]

**Rearranging:**

\[ r \ = \ \dfrac{ a_n }{ a_{ n – 1 } } \]

## Explanation of Common Ratio Using Examples

### Spreading of a Virus

The spreading of a **virus or a bacterial infection** in a population is an **example of geometric progression**. Whenever a virus outbreak occurs in a population, the **scientists define the R value** of the virus. This R value tells us how many other people can get infected from a single person. This **R value is nothing but the common ratio**.

Consider the case of **coronavirus pandemic**. The R value or the **common ratio** of this virus is around **2.5**. This means that if one of the persons gets infected, it will spread it to 2.5 others. We can **construct a geometric sequence** using 1 as the first number and the value 2.5 as **common ratio:**

{ 1, 2.5, 6.25, 15.63, 39.06, …… }

The following figure shows a **chart of this progression**.

**Figure 2: Common Ratio in Spreading of Coronavirus Outbreak**

### Nuclear Chain Reaction

Another very interesting example can be the **nuclear fission chain reaction** used in the nuclear power plants. Whenever an atom splits, its called **nuclear fission.** Each fission event is caused by the collision of a **high speed neutron**. The fission reaction may release **two new high speed neutrons** that further collide with the other nuclei and create a **chain reaction.** This process can also be modeled using the **geometric progression** and **common ratio**.

Lets say that their is only** one high speed neutron at the start** $ a_1 \ = \ 1 $ and **each fission reaction releases two neutrons** $ r \ = \ 2 $. We can construct the sequence of no. of fission reactions as follows:

{ 1, 2, 4, 8, 16, …… }

This process can be shown in the form of a graph as follows:

**Figure 3: Common Ratio in Nuclear Chain Reaction**

## Use of Logarithmic Scales in Common Ratio

The **geometric progressions are exponential,** hence its very useful to use **logarithmic scales for their graphs**. Consider the following figure where the spreading of coronavirus is **plotted over long term**.

**Figure 4: Logarithmic Scale for Common Ratio and Geometric Sequences**

It can be clearly seen that the **logarithmic scale** **is** **more readable** over long-term data. Logarithms are often used in the calculation of common ratios for **numerical problems** of complex nature.

## A Numerical Example of Common Ratio

**Part (a):** The **3rd and 4th elements** of a geometric progression are **2.25 and 3.375** respectively. Find the **common ratio**.

**Part (b):** The **5th element** of a geometric progression is **625** and its **common ratio is 5**, then calculate the value of **first element**.

### Part (a) Solution

**Given that:**

\[ { a_{ n – 1 } } \ = \ 2.25 \]

\[ { a_n } \ = \ 3.375 \]

**Using the formula:**

\[ r \ = \ \dfrac{ a_n }{ a_{ n – 1 } } \]

r = 3.375 / 2.25

**r = 1.5**

Which is the **value of common ratio.**

### Part (b) Solution

**Given that:**

r = 5

n = 5

\[ a_n \ = \ 625 \]

**Using the formula:**

\[ a_n \ = \ a_1 \ r^{ n – 1 } \]

\[ 625 \ = \ a_1 \ ( 5 )^{ 5 – 1 } \]

\[ 625 \ = \ a_1 \ ( 5 )^{ 4 } \]

\[ 625 \ = \ a_1 \ (125) \]

**Rearranging** the above expression,

\[ a_1 \ = \ \dfrac{ 625 }{ 125 } \]

\[ a_1 \ = \ 1 \]

Which is the **first value of the sequence**.

*All figures and graphs have been constructed using GeoGebra.*