Geometric Progression (GP): Definition, Formula, Nth Term and Sum

Understanding geometric progression is crucial for students preparing for competitive and placement interviews. It showcases a deep understanding of mathematical patterns and growth, which can impress interviewers.

Students can demonstrate their analytical skills and problem-solving abilities by grasping concepts like the ratio between consecutive terms and applying them to real-world situations.

What is Geometric Progression?

A geometric progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio.

Mathematically, a geometric progression looks like this:

a,ar,ar²,ar³,…

• a is the first term,
• r is the common ratio (a fixed number),
• Each term is obtained by multiplying the previous term by r.

For example, in the sequence 2, 6, 18, 54,…, the common ratio is 3 because each term is 3 times the previous one.

Key Properties of Geometric Progression

Let us consider some of the important properties of geometric progression:

Constant Ratio

Each term is obtained by multiplying the previous term by a constant value called the common ratio ($r$). This ratio remains the same throughout the sequence.

r=T_n+1/T_n​​

General Term

The $n$th term (T_n) of a GP can be found using the formula:

T_n=a×rⁿ⁻¹

• a is the first term,
• r is the common ratio,
• and n is the term number.

Product of Two Terms

The product of any two terms equidistant from the start and end of a finite GP is the same. If you have a GP with terms T₁,T₂,…,T_n then:

T_k×T_n−k+1=a×T_n​

This holds for all k.

Sum of First $n$ Terms (Finite GP)

The sum of the first $n$ terms of a geometric progression is given by:

• If r≠1r, then S_n=a(1−rⁿ)/1−r​
• If r=1, then S_n=n×a

Sum of Infinite GP (only if ∣r∣<1|)

For an infinite geometric progression, the sum is given by:

S_∞=a/1−r​

This only applies if the absolute value of the common ratio ∣r∣ is less than 1.

Non-zero Terms

In a geometric progression, none of the terms can be zero (since dividing or multiplying by zero would disrupt the sequence).

Exponential Growth or Decay

Depending on whether the common ratio $r$ is greater than 1 (growth) or between 0 and 1 (decay), the sequence either increases or decreases exponentially.

General Form Of Geometric Progression 

The general form of a geometric progression can be expressed as a, ar, ar², ar³, ..., ar^(n-1), where 'a' represents the first term, and 'r' denotes the common ratio.

The sequence starts with 'a', and each subsequent term is created by multiplying the previous term by 'r'. This consistent multiplication creates a pattern where the terms increase or decrease at a fixed rate determined by the common ratio.

Common Ratio

The common ratio plays a crucial role in establishing the relationship between consecutive terms in a geometric progression. It dictates how each term is related to its preceding one. If 'r' is greater than 1, the terms will increase exponentially. Conversely, if 'r' is between 0 and 1, the terms will decrease gradually.

For instance, in a GP with a common ratio of 2 starting from 1 (1, 2, 4, 8, ...), each term is twice the preceding one. This showcases how the common ratio influences growth or decline during progression.

General Term or the Nth Term of Geometric Progression

The general term or $n$th term of a geometric progression (GP) is the formula used to find any specific term in the sequence without having to list all the preceding terms.

The formula is T_n=a×rⁿ⁻¹

• T_nis the nth term,
• a is the first term of the GP,
• r is the common ratio,
• n is the position of the term in the sequence.

Example of Nth Term

For the geometric progression 3,6,12,24,…

• First term a=3
• Common ratio r=6/3=2

To find the 5th term (T₅):

T₅=3×2⁵⁻¹=3×2⁴=3×16=48

The sum of nth Terms of GP

To find the sum of the first 'n' terms of a geometric progression (GP), you can use these formulas based on the common ratio '$r\text{'}$:

1. When r≠1: The sum S_n​ of the first '$n\text{'}$ terms of a GP is given by: S_n=a(rⁿ⁻¹)/r−1​

• a is the first term,
• r is the common ratio,
• n is the number of terms.

2. When r=1: If the common ratio $r$ is $1$, the formula simplifies to S_n=n×a

Examples

1. For r≠1:

   • Sequence: 2,6,18,54,…
   • First term a=2, common ratio r=3, and n=4 terms.

   Sum S₄ calculated as:

   S₄=2(3⁴⁻¹)/3−1=2(81−1)/2=2×80/2=80

2. For r=1:

   • Sequence: 5,5,5,5,…
   • First term a=5, common ratio r=1, and n=4 terms.

   Sum S₄ is calculated as:

   S₄=4×5=20

Types Of Geometric Progression

Click here to enhance and upskill your quantitative aptitude by exploring different topics, including geometric progression.

Solved Questions and Answers of GP 

Practice is the key to excelling in quantity aptitudes, such as geometric progression! Below are some of the top selected practice questions with answers:

Question 1: What is the fifth term of the geometric progression 2, 4, 8, 16, ...?

a) 23

b) 43

c) 18

d) 32

Solution: d) 32

Explanation: T_n​=a×r⁽ⁿ⁻¹⁾

• T_n​ is the nth term of the progression

• a is the first term of the progression

• r is the common ratio of the progression

• First term (a): 2

• Common ratio (r): 4 ÷ 2 = 2

• n: 5

Substituting the values into the formula:

T₅​=2×2⁽⁵⁻¹⁾= 2×2⁴= 2×16

T₅​=32

So, the fifth term of the geometric progression is 32.

Question 2: If the first term of a geometric progression is 3 and the common ratio is 2, what is the fourth term?

a) 24

b) 18

c) 20

d) 32

Solution: a) 24

Explanation: Follow the steps as provided in question 1, and plug these values into the formula to solve the question to get the answer: r=2, a=3, n=4.

Question 3: What is the common ratio of the geometric progression 1, 2, 4, 8, ...?

a) 3

b) 2

c) 4

d) 5

Solution: b) 2

Explanation: Divide the n^th term by the (n - 1)^th term

Question 4: If the third term of a geometric progression is 16 and the common ratio is 2, what is the first term?

a) 4

b) 6

c) 8

d) 2

Solution: a) 4

Explanation: Plugging these values a=16, r=2, n=3 into the formula, T_n​=a×r⁽ⁿ⁻¹⁾ will give you the answer. 

Question 5: What is the sum of the first 5 terms of the geometric progression 3, 6, 12, 24, ...?

a) 90

b) 86

c) 93

d) 88

Solution: c) 93

Explanation: Plugging these values a=3, r=2, and n=5 into the formula S_n = a[(r^{n – 1})/(r – 1)]. It will provide the correct answer as 93. 

Question 6: If the sum of the first three terms of a geometric progression is 42 and the common ratio is 2, what is the first term?

a) 7

b) 8

c) 9

d) 6

Solution: d) 6

Explanation: Plugging these values S_n=42, r=2, n=3 into the formula S_n = a[(r^{n – 1})/(r – 1)]. It will provide the correct answer as 6. 

Question 7: What is the seventh term of the geometric progression 5, 10, 20, 40, ...?

a) 320

b) 220

c) 380

d) 423

Solution: a) 320

Explanation: T_n = a×r^n-1
r=2, a=5, n=7

Question 8: If the third term of a geometric progression is 27 and the fifth term is 243, what is the common ratio?

a) 6

b) 3

c) 8

d) 9

Solution: 3

Explanation: T_n = a×r^n-1
T₅/T₃

Question 9: Find the sum of the first 10 terms of the geometric progression 2, 6, 18, 54, ...?

a) 59048

b) 49152

c) 53248

d) 57344

Solution: a) 59048

Explanation: S_n = a[(r^{n – 1})/(r – 1)]
r=3, n=10, a=2

Question 10: If the sum of the first four terms of a geometric progression is 30 and the common ratio is 3, what is the first term?

a) 2.5

b) 0.75

c) 5

d) 1.875

Solution: b) 0.75

Explanation: S_n= a[(r^{n – 1})/(r – 1)]
S_n=30, r=3, n=4

Conclusion

The comprehensive exploration of geometric progression has shed light on its fundamental concepts, properties, and applications. One can grasp the essence of this mathematical sequence by delving into its general form, common ratio, types, Nth term explanation, and summing techniques.

The insights gained from analyzing finite and infinite sums enhance your mathematical proficiency and equip you with a powerful tool for analyzing patterns and predicting outcomes in diverse contexts.

Frequently Asked Questions (FAQs)

1. What is a geometric progression?

In geometric progression, a series of numbers is created by multiplying each successive term by a constant, non-zero value known as the common ratio.

2. How is the Nth term of a geometric progression calculated?

The nth term of a GP can be calculated using the formula Tₙ = a×r^n-1, where (Tₙ) represents the nth term. The initial term is denoted by (a), the unchanging ratio is represented as (r), and (n) indicates the term's position.

3. What does summing finite GP terms refer to?

Summing finite GP terms refers to finding the total sum of a specific number of terms in a geometric progression. 

4. How can geometric progressions be applied in real-life scenarios?

Geometric progressions find applications in various fields, such as compound interest calculations, population growth models, computer algorithms (searching techniques), and physics (decay processes). Understanding geometric progressions aids in efficiently analyzing exponential growth or decay situations.

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