An introduction to Geometric Progression

The geometric progression is a sequence of numbers that follows a special pattern. The geometric progression is abbreviated as GP. In this section, we will learn about geometric progression.

What is a Geometric Progression (GP)

A geometric progression is a special type of sequence of non-zero numbers where each term (except the first term) is determined by multiplying its preceding term with a fixed non-zero constant quantity. The fixed constant quantity is called the common ratio of the GP.

For example, $1, 2 , 4, 8, \cdots$ is a geometric progression as every term is non-zero and the common ratio is $2$ as $\frac{2}{1}$ $=\frac{4}{2}$ $=\cdots$ $=2.$

 

Terms and Notations

In a geometric progression, the following notations are generally used to denote important terms of the GP.

$a$: the first term

$r$: the common term

$n$: the number of terms

$a_n$: the n-th term

$S_n$: the sum of the first n terms

 

General Form and n-th term of a Geometric Progression

Let us consider a geometric progression with the first term $a$ and with the common ratio $r.$ Form the above definition of a GP, we get each term by multiplying the preceding term with $r.$ Therefore, we have:

The first term $=a$ $=ar^{1-1}$

The second term $=a \times r$ $=ar$ $=ar^{2-1}$

The third term $=ar \times r$ $=ar^2$ $=ar^{3-1}$

The fourth term $=ar^2 \times r$ $=ar^3$ $=ar^{4-1}$

So we can say that the n-th term of the GP is $a_n=ar^{n-1}.$ Hence the general form of a geometric progression with the first term $a$ and with the common ratio $r$ is given as follows: $a, ar,$ $ar^2,$ $ar^3 \cdots$

 

Examples of Geometric Progressions

(i) $1, 5, 25, 125, \cdots$ is a geometric progression as every term is non-zero and the common ratio is $r=5$. This GP has the first term $5.$

(ii) $1, -3, 9, -27, \cdots$ is an example of a GP with the negative common ratio $r=-3.$ Each term is non-zero and the first term of the GP is $1.$

(iii) In a similar way, $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \cdots$ is a geometric progression with common ratio $r=\frac{1}{2}$. The common ratio of the GP is a fraction with the first term $\frac{1}{2}.$

Remark: From the above examples, we see that the common ratio of a geometric progression can be either positive, negative, or fraction. But it can never be zero.

 

Geometric Series

Let $a_1, a_2, \cdots, a_n, \cdots$ be a geometric progression. Then the sum $a_1+a_2+\cdots+a_n+\cdots$ $=\sum_{n=0} a_n$ is called the geometric series of the given GP.

If the GP contains a finite number of terms, then the series is called a finite geometric series; otherwise, it is called an infinite geometric series.

 

Finite Geometric Progression

Let us consider a finite geometric progression of $n$ elements. If the GP has the first term $a$ and the common ratio $r$, then the sum of the terms of the GP is given as follows:

$S=\frac{a(1-r^n)}{1-r}$   when $-1<r<1$

$\quad=\frac{a(1-r^n)}{1-r}$   when $r<-1$ or $r>1$

$\quad=na$   when $r=1$

 

Infinite Geometric Progression

If an infinite geometric progression has the first term $a$ and the common ratio $r$ with $-1<r<1$, then the value of the infinite geometric series is $\frac{a}{1-r}$. More precisely,

$a+ar+ar^2+\cdots=\frac{a}{1-r}$  if $-1<r<1.$

A special case where $a=1$, we have

$1+r+r^2+\cdots=\frac{1}{1-r}$  if $-1<r<1.$

 

Geometric Mean

The geometric mean (GM) of the two numbers $a$ and $b$ is given by the following quantity:

GM $=\sqrt{ab}$

Note that if $z$ is the geometric mean of $a$ and $b$, then we must have that the three numbers $a, z$ and $b$ form a GP.

The geometric mean of n numbers $a_1, a_2, \cdots, a_n$ is given by the n-th root of the product of the given numbers. More precisely, in this case, we have

GM $=\sqrt[n]{a_1 a_2 \cdots a_n}.$

 

Formulas of Geometric Progressions

For a geometric progression (GP) having the first term $a$ and the common ratio $r$, we have the following results of the GP.

• The GP has the form $a, ar, ar^2, \cdots$

• The n-th term $a_n=ar^{n-1}$

• The sum of the first n terms is given as follows:

$S$ $=\frac{a(1-r^n)}{1-r}$ if $-1<r<1$

and

$S$ $=\frac{a(r^n-1)}{r-1}$ if $r<-1$ or $r>1$

• The geometric mean of $a$ and $b$ is $\sqrt{ab}$

• If the three numbers are in a GP, then we should assume the numbers as follows: $\frac{a}{r},$ $a$ and $ar.$

 

Solved Problems of Geometric Progression

Problem 1: Find few terms of a geometric progression with first term $2$ and common ratio $\frac{1}{3}$

Solution:

The first term is $a=2$ and the common ration is $r=\frac{1}{3}.$ So the second term $=2 \times \frac{1}{3}$ $=\frac{2}{3}$, the third term $=\frac{2}{3} \times \frac{1}{3}$ $=\frac{2}{9}$, the fourth term $=\frac{2}{9} \times \frac{1}{3}$ $=\frac{2}{27}.$ So the few term of the GP are \[2, \frac{2}{3}, \frac{2}{9}, \frac{2}{27} \cdots\]