what is geometric progression? what are its formulas?

 

Introduction

In the world of mathematics, geometric progression holds a significant place due to

its widespread application across various fields. Whether it's financial planning,

population growth analysis, or even artistic design, understanding the principles of

geometric progression can provide valuable insights and open up a world of

possibilities. In this article, we will explore the concept of geometric progression, its

properties, and its real-world application

What is a geometric progression?

Geometric progression, also known as a geometric sequence, is a sequence

of numbers in which each term after the first is found by multiplying the

previous term by a fixed, non-zero number called the common ratio (r).

Mathematically, a geometric progression can be represented as:

a, ar, ar², ar³, ...

Here,

'a' is the first term,

'r' is the common ratio, and

'n' represents the number of terms in the sequence.

The power of geometric progression lies in its predictable pattern, where

each term is a product of the previous term and the common ratio.

What are the properties of Geometric Progression?

Geometric progressions possess several unique properties that make them

fascinating and useful:

1. Common Ratio: The common ratio (r) determines the relationship between

consecutive terms in the progression.

2. If the common ratio is greater than 1, the progression is increasing, while a

common ratio between 0 and 1 produces a decreasing progression.

Formula of Geometric progression

Formula of Geometric progression is given by,

General Form of Geometric Progression

The general form of Geometric Progression is:

a, ar, ar2, ar3, ar4,…, arn-1

                          Where,

                                   a = First term

                                   r = common ratio

                                   arn-1 = nth term

General Term or Nth Term of Geometric Progression

Nth term of Geometric progression is given by:

Let a be the first term and r be the common ratio for a Geometric Sequence.

                Then, the second term, a2 = a × r = ar

                                Third term, a3 = a2 × r = ar × r = ar2

                                 nth term, an = arn-1

                Therefore, the formula to find the nth term of GP is:

          Note: The nth term is the last term of finite GP

Formula for nth Term:

The formula for finding the nth term of a geometric progression is given by:

 an  = arn-1

This formula allows us to calculate any term in the sequence without the need to

list all the preceding terms.

General Formula For Finding First' n ' Terms

The sum of the first 'n' terms of a geometric progression can be calculated using the

formula:

Sn = a * (1 -rn ) / (1 - r)

This formula helps in determining the total value or quantity represented by the

sequence.

some problems of Geometric progression

Question  

If the first term of a G.P. is 20 and the common ratio is 4. Find the 5th term.

Solution

Given,

First term, a=20

Common ratio, r=4

We know,

Nth term of G.P.,

an = arn-1

⇒ a5 = 20×44

= 20×256

= 5120

Question

The sum of the first three terms of a G.P. is 21/2 and their product is 27. Find the common ratio.

Solution

 Let three terms of G.P. be a/r, a, ar.

Given,

Product of first three terms = 27

⇒ (a/r) (a) (ar) = 27

⇒ a3 = 27

⇒ a = 3.

Sum of first three terms = 21/2

⇒ (a / r + a + ar) = 21/2

⇒ a (1 / r + 1 + 1r) = 21/2

⇒ (1 / r + 1 + 1r) = (21/2)/3 = 7/2

⇒ (r2 + r + 1) = (7/2) r

⇒ r2 – (5/2) r + 1 = 0

⇒ r = 2 and ½

Question

 Find a Geometric Progress for which the sum of first two terms is -4 and the fifth term is 4 times the third term.

Solution

 Let the first term of the geometric series be a and the common ratio is r.

Sum of the first two terms = -4

a + ar = -4 ………………(i)

Fifth term is 4 times the third term.

ar4 = 4ar2

r2 = 4

r = ±2

If we consider r = 2, then putting value of r in eq.(i)

a(1+2) = -4

a = -4/3

ar = -8/3

ar2 = -16/3

Thus, the G.P. is -4/3, -8/3, -16/3, …..

Question

 The number 2048 is which term in the following Geometric sequence 2, 8, 32, 128, . . . . . . . . .

Solution

 Here a = 2 and r = 4

nth term G.P is an = arn-1

⇒ 2048 = 2 x ( 4) n-1

⇒ 1024 =( 4) n-1

⇒ ( 4) 5 = ( 4) n-1

⇒ n = 6

Question

 In a G.P, the 6th term is 24 and the 13th term is 3/16 then find the 20th term of the sequence.

Solution

 Let first term be ‘a’ and common ratio is ‘r’

Given,

a6 = 24 ————- ( i)

a13 = 3/16 ————– ( ii)

⇒ a6 = a r6-1

a13 = a r13-1

⇒ 24 = a r5

3/16 = a r12

⇒ r7 = 3 / 24 x 16 = 1 / (2)7

⇒ r = 1/2 ———– (iii)

Thus,

⇒ a6 = 24 = a (1/2)5

⇒ a = 3 x 28

Now a20 = a r20-1

a20 = 3 x 28 x ( 1/2 )19 = 3 / 211

Question

Find the sum of the geometric series:

4 – 12 + 36 – 108 + ………….. to 10 terms

Solution

 The first term of the given Geometric Progression = a = 4 and its common ratio = r = −12/4 = -3.

Sum of the first 10 terms of geometric series:

S10 = a. (rn – 1/r-1)

= 4. ((-3)10 – 1)/(-3-1)

= – (-3)10 – 1

= – 59048

Question

 ‘x’ and ‘y’ are two numbers whose AM is 25 and GM is 7. Find the numbers.

Solution

 Here x’ and ‘y’ are two numbers then

Arithmetic mean = AM = (x+y)/2

25 = (x+y)/2

x+y = 50 ………(i)

Geometric mean, GM = √(xy)

7 = √(xy

72 = xy

xy = 49 ………..(ii)

Solving equation (i) and (ii), we get;

x = 1 and y = 49.

Question

Determine the common ratio r of a geometric progression with the first term is 5 and fourth term is 40.

Solution

Given,

First term, a1 = 5

Fourth term, a4 = 40

a4/a1 = 40/5

a1r3/a1 = 40/5

r3 = 8

r = 2

Question

If the nth term of a GP is 128 and both the first term a and the common ratio r are 2. Find the number of terms in the GP.

Solution

 nth term of a GP, an = 128

First term of GP, a = 2

Common ratio, r = 2

Nth term of G.P., an = a.rn-1

128 = 2.2n-1

64 = 2n-1

26 = 2n-1

n- 1 = 6

n = 7

Therefore, there are 7 terms in GP.

Question

What is the sum of infinite geometric series with first term equal to 1 and common ratio is ½?

Solution

By the formula of sum of infinite geometric series, we have;

S = a1 1/(1 – r)

S = 1. 1/(1-½)

S = 1/(½)

S = 2

Hence, the required sum is 2.

Applications in Real Life

Geometric progressions find numerous applications in various fields, including:

• Finance and Investment: Geometric progressions are widely used in compound interest calculations. For instance, calculating the growth of investments over time, where the interest is compounded annually or at regular intervals.
• Population Growth: Geometric progressions help model population growth by considering birth and death rates. It provides insights into predicting future population trends, resource allocation, and urban planning.
• Art and Design: Artists and designers often utilize geometric progressions to create visually appealing patterns, such as the famous Fibonacci sequence. By applying geometric progressions, they can create aesthetically pleasing and harmonious compositions.
• Computer Science: Geometric progressions are employed in various algorithms, data structures, and cryptography. For example, in binary search algorithms and encryption techniques, the concept of geometric progression plays a crucial role.

SUMMARY

Geometric progression serves as a fundamental concept in mathematics and finds

extensive applications in multiple domains. From financial planning to population

modeling, this powerful mathematical tool helps us understand and predict patterns

in the real world. By harnessing the properties and formulas associated with

geometric progression, individuals and professionals can unlock new insights,

make informed decisions, and pave their path to success.

So, whether you're an investor, a scientist, an artist, or a computer programmer,

understanding geometric progression opens doors to endless possibilities and

enhances your problem-solving abilities. Embrace the power of geometric

progression and embark on a journey of discovery and growth.