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Geometric Progression (GP) | Formula and Properties

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Geometric Progression (GP) is a sequence of numbers where each next term in the progression is produced by multiplying the previous term by a fixed number. The fixed number is called the Common Ratio.

Geometric Progression

 Let’s learn the formulas and properties of Geometric Progression with the help of solved examples.

What is Geometric Sequence?

Geometric sequence is a series of numbers in which the ratio between two consecutive terms is constant. This ratio is known as the common ratio denoted by ‘r’, where r ≠ 0.

The nth term of Geometric series is denoted by an and the elements of the sequence are written as a1, a2, a3, a4, …, an.

Condition for the given sequence to b a geometric sequence :

a2/a1 = a3/a2 =  … = an/an-1 = r (common ratio)

Geometric Progression Formula

The list of formulas related to GP is given below :

 

Formulas

 

General Form

a,ar,ar2,ar3,…

a is the first term, r is the common ratio.

nth Term of a GP

Tn = arn-1

Tn is the nth term, a is the first term, r is the common ratio.

Common Ratio

r = Tn/ Tn-1​​

Tn​ and Tn-1​ are consecutive terms of the GP.

Sum of First n Terms (r > 1)

Sn = a[(rn – 1)/(r – 1)]

Sn​ is the sum of first n terms, and >1r>1.

Sum of First n Terms (r < 1)

Sn = a[(1 – rn)/(1 – r)]

Sn is the sum of first n terms, and <1r<1.

nth Term from End (finite GP)

r = l/ [r(n – 1)]

l is the last term, n is the term position from the end.

Sum of Infinite GP

Valid only if 0 < r < 1.

S= a/(1 – r) ​

Geometric Mean

b=ac

For three quantities a,b,c in GP, b is the geometric mean of a andc.

kth Term from End (finite GP)

Tk = arn-k.

Tk is the kth term from the end, n is the total number of terms.

General Form of Geometric Progression

The given sequence can also be written as:

a, ar, ar2, ar3, … , arn-1  

Here, r is the common ratio and a is the scale factor

Common ratio of Geometric Series is given by:

r = successive term/preceding term = arn-1 / arn-2

Nth Term of Geometric Progression

The terms of a GP are represented as a1, a2, a3, a4, …, an.

Expressing all these terms according to the first term a1, we get

a1 = a
a2 = a1r
a3 = a2r = (a1r)r = a1r2
a4 = a3r = (a1r2)r = a1r3

am = a1rm−1

Similarly,
an = a1rn – 1

General term or nth term of a Geometric Sequence a, ar, ar2, ar3, ar4 is given by : 

an = arn-1

where, 

a1 = first term, 
a2 = second term
an = last term (or the nth term)

Nth Term from the Last Term is given by:

an = l/rn-1

where,
l is the last term

Sum of N Terms of GP

The sum of geometric progression is given by

S = a1 + a2 + a3 + … + an

S = a1 + a1r + a1r2 + a1r3 + … + a1rn−1     ….equation (1)

Multiply both sides of Equation (1) by r (common ratio), and we get

S × r= a1r + a1r2 + a1r3 + a1r4 + … + a1rn     ….equation (2)

Subtract Equation (2) from Equation (1)

S – Sr = a1 – a1rn

(1 – r)S = a1(1 – rn)

Sn = a1(1 – rn)/(1 – r), if r<1

Now, Subtracting Equation (1) from Equation (2) will give

Sr – S = a1rn a1

(r – 1)S = a1(rn-1)

Hence, Sum of First n Terms of GP is given by:

Sn = a(1 – rn)/(1 – r), if r < 1

Sn = a(rn -1)/(r – 1), if r > 1

Sum of Infinite Geometric Progression

The number of terms in infinite geometric progression will approach infinity (n = ∞). The sum of infinite geometric progression can only be defined at the range of |r| < 1.

Let us take a geometric sequence a, ar, ar2, … which has infinite terms. S denotes the sum of infinite terms of that sequence, then

S = a + ar + ar2 + ar3+ … + arn +..(1)

Multiply both sides by r,

rS = ar + ar2 + ar3+ … … (2)

subtracting eq (2) from eq (1),

S – rS = a

S (1 – r) = a

Thus, Sum of Infinite Geometric Progression is given by,

S= a/(1-r), where |r| < 1

Properties of Geometric Progression

Geometric Sequence has the following key properties :

  • a2k = ak-1 × ak+1
  • a1 × an = a2 × an-1 =…= ak × an-k+1
  • If we multiply or divide a non-zero quantity by each term of the GP, then the resulting sequence is also in GP with the same common difference.
  • Reciprocal of all the terms in GP also forms 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.

Types of Geometric Progression

GP is further classified into two types, which are:

  1. Finite Geometric Progression (Finite GP)
  2. Infinite Geometric Progression (Infinite GP)

Finite Geometric Progression

Finite G.P. is a sequence that contains finite terms in a sequence and can be written as a, ar, ar2, ar3,……arn-1, arn

An example of Finite GP is 1, 2, 4, 8, 16,……512

Infinite Geometric Progression

Infinite G.P. is a sequence that contains infinite terms in a sequence and can be written as a, ar, ar2, ar3,……arn-1, arn……, i.e. it is a sequence that never ends.

Examples of Infinite GP are:

  • 1, 2, 4, 8, 16,……..
  • 1, 1/2, 1/4, 1/8, 1/16,………

Geometric Sequence Recursive Formula

A recursive formula defines the terms of a sequence in relation to the previous value. As opposed to an explicit formula, which defines it in relation to the term number.

For an example, let’s look at the sequence: 1, 2, 4, 8, 16, 32

Recursive formula of Geometric Series is given by

term(n) = term(n – 1) × 2

In order to find any term, we must know the previous one. Each term is the product of the common ratio and the previous term.

term(n) = term(n – 1) × r

Example: Write a recursive formula for the following geometric sequence: 8, 12, 18, 27, … 

Solution: 

The first term is given as 6. The common ratio can be found by dividing the second term by the first term.

r = 12/8 = 1.5

Substitute the common ratio into the recursive formula for geometric sequences and define  a1

term(n) = term(n – 1) × r 

= term(n -1) × 1.5 for n>=2

a1 = 6

Read More :

Geometric Progression vs Arithmetic Progression

Here are the key differences between Geometric Progression and Arithmetic Progression :

Difference between Arithmetic Sequence and Geometric Sequence

 

Arithmetic Sequence

Geometric Sequence

Definition

A sequence in which the difference between any two consecutive terms is constant.

A sequence in which the ratio of any two consecutive terms is constant.

Common Term

The common difference, denoted as ‘d’.

The common ratio, denoted as ‘r’.

General Formula

The nth term is given by an​=a1+(n−1)d, where a1​ is the first term and ‘d’ is the common difference.

The nth term is given by, an​=a1×r(n−1), where a1​ is the first term and ‘r’ is the common ratio.

Example

2, 5, 8, 11, 14, … (Here, d = 3)

3, 6, 12, 24, 48, … (Here, r = 2)

Nature of Growth

Linear growth: The terms increase or decrease by a constant amount.Exponential growth: The terms increase or decrease by a constant factor.

Graph Appearance

Forms a straight line when plotted on a graph.Forms a curve (exponential growth or decay) when plotted on a graph.

Sum of n Terms

Given by Sn​= n​/2[2a1 +(n−1)d]

Given by Sn = a1(rn -1)/(r – 1)

Related :

Solved Examples on Geometric Sequence

Let’s solve some example problems on Geometric sequence.

Example 1: Suppose the first term of a GP is 4 and the common ratio is 5, then the first five terms of GP are?

Solution: 

First term, a = 4
Common ratio, r = 5
Now, the first five term of GP is
a, ar, ar2, ar3, ar4
a = 4
ar = 4 × 5 = 20
ar2 = 4 × 25 = 100
ar3 = 4 × 125 = 500
ar4 = 4 × 625 = 2500
Thus, the first five terms of GP with first term 4 and common ratio 5 are:
4, 20, 100, 500, and 2500

Example 2: Find the sum of GP: 1, 2, 4, 8, and 16.

Solution: 

Given GP is 1, 2, 4, 8 and 16
First term, a = 1
Common ratio, r = 2/1 = 2 > 1
Number of terms, n = 5
Sum of GP is given by;
Sn = a[(rn – 1)/(r – 1)]
S5 = 1[(25 – 1)/(2 – 1)]
     = 1[(32 – 1)/1]
     = 1[31/2]
     = 1 × 15.5
     = 15.5

Example 3: If 3, 9, 27,…., is the GP, then find its 9th term.

Solution: 

nth term of GP is given by:

an = arn-1

given, GP 3, 9, 27,….
Here, a = 3 and r = 9/3 = 3
Therefore,
a9 = 3 x 39 – 1
    = 3 × 6561
    = 19683

Geometric Progression- FAQs

What is Geometric Progression Definition?

Geometric Progression (GP) is a specific type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed constant, which is termed a common ratio(r).

For example, 1, 3, 9, 27, 81, …….

What is the common ratio of GP?

Common multiple between each successive term in a GP is termed the common ratio. It is a constant that is multiplied by each term to get the next term in the GP. If a is the first term and ar is the next term, then the common ratio is equal to:
ar/a = r

Write the general form of GP.

General form of a Geometric Progression (GP) is a, ar, ar2, ar3, ar4,…,arn-1

a = First term
r = common ratio
arn-1 = nth term

What is sum of n terms of GP formula?

The formula to find the sum of GP is:

Sn = a + ar + ar2 + ar3 +…+ arn-1

Sn = a[(rn – 1)/(r – 1)]

where r ≠ 1 and r > 1

What is Geometric Progression sum to infinity ?

Formula for the sum to infinity of a geometric series is:

S = a / (1 – r)

where:

  • S​ is the sum to infinity.
  • a is the first term of the series.
  • r is the common ratio of the series.


Last Updated : 11 Jan, 2024
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