Sequence & Series – GP & Sum of GP | Class 11 Maths + JEE Main 2027
Geometric Progression (GP) is an important topic in Class 11 Mathematics and JEE Main. It is used to calculate the nth term of a sequence, the sum of a finite number of terms, and the sum of an infinite series under specific conditions.
If you understand the common ratio and remember the correct formulas, many GP questions become straightforward. In this article, we will cover GP basics, nth-term formulas, finite and infinite sums, geometric mean, important properties, solved examples, and common mistakes to avoid in JEE Main.
1. What Is Geometric Progression (GP)?
Geometric Progression ek aisi sequence hai jisme consecutive terms ka common ratio constant hota hai. In simple words, every term is obtained by multiplying the previous term by the same fixed number. This number is called the common ratio, represented by r.
Example of GP
2, 6, 18, 54, 162, ...
Calculate the ratio of consecutive terms:
r = 6/2 = 3
r = 18/6 = 3
r = 54/18 = 3
Since the ratio remains constant, the sequence is a Geometric Progression with common ratio 3.
General Form of GP
a, ar, ar², ar³, ar⁴, ...
Where:
- a = First term
- r = Common ratio
- n = Number of terms
How to Identify a GP Quickly
Divide each term by the previous term. If the ratio is the same throughout the sequence, it is a GP.
For example:
- 3, 6, 12, 24, ... → GP with r = 2
- 5, 15, 45, 135, ... → GP with r = 3
- 2, −4, 8, −16, ... → GP with r = −2
Quick Trick: Consecutive terms ka ratio same hai, toh sequence GP hai. Remember that the common ratio can be positive, negative, or fractional.
2. Nth Term of a GP
The nth-term formula helps us find a particular term without writing all the terms before it. For JEE Main, this formula is especially useful when the question asks for the 10th, 15th, or any other term of a GP.
Formula for the Nth Term
aₙ = arⁿ⁻¹
Where:
- aₙ = nth term
- a = First term
- r = Common ratio
- n = Position of the required term
Why Is the Exponent n − 1?
Consider the GP:
a, ar, ar², ar³, ar⁴, ...
- First term: a₁ = a = ar⁰
- Second term: a₂ = ar
- Third term: a₃ = ar²
- Fourth term: a₄ = ar³
The exponent is always one less than the term number. Therefore, the nth term is arⁿ⁻¹.
Solved Example: Find the 6th Term
Consider the GP:
3, 6, 12, 24, ...
Given:
- a = 3
- r = 2
- n = 6
Apply the formula:
aₙ = arⁿ⁻¹
a₆ = 3 × 2⁶⁻¹
a₆ = 3 × 2⁵
a₆ = 3 × 32 = 96
Final Answer: 96
JEE Main Tip: Nth-term formula mein exponent n − 1 hota hai, n nahi. This is one of the most common mistakes in GP questions.
3. Sum of the First n Terms of a GP
If a question asks for the sum of the first n terms of a GP, use the finite-sum formula. The formula depends on whether the common ratio is equal to 1 or not.
Case 1: When r ≠ 1
The sum of the first n terms is:
Sₙ = a(rⁿ − 1)/(r − 1)
An equivalent form is:
Sₙ = a(1 − rⁿ)/(1 − r)
Both formulas give the same answer. Choose the form that makes calculations easier.
Case 2: When r = 1
If the common ratio is 1, every term is equal to the first term:
a, a, a, a, ...
Therefore, the sum of n terms is:
Sₙ = na
Do not substitute r = 1 directly into the general finite-sum formula because its denominator becomes zero. Use Sₙ = na instead.
Solved Example: Sum of a GP
Find the sum:
2 + 4 + 8 + 16 + 32
Given:
- a = 2
- r = 2
- n = 5
Use the formula:
Sₙ = a(rⁿ − 1)/(r − 1)
S₅ = 2(2⁵ − 1)/(2 − 1)
S₅ = 2(32 − 1)/1
S₅ = 2 × 31
S₅ = 62
Final Answer: 62
Quick Verification
Add the terms directly:
2 + 4 + 8 + 16 + 32 = 62
The answer matches the formula. In an examination, the formula is especially useful when n is large and adding every term individually would take too much time.
4. Sum of an Infinite GP
An infinite GP contains infinitely many terms. However, its infinite sum can have a finite value if the terms become sufficiently small and approach zero.
Condition for an Infinite GP to Have a Finite Sum
|r| < 1
The absolute value of the common ratio must be less than 1. This means the common ratio lies between −1 and 1.
Formula for the Infinite Sum
S∞ = a/(1 − r)
Where:
- S∞ = Sum of the infinite GP
- a = First term
- r = Common ratio, with |r| < 1
Solved Example: Find the Infinite Sum
Find the sum:
8 + 4 + 2 + 1 + ...
Identify the values:
- a = 8
- r = 4/8 = 1/2
Since |1/2| < 1, the infinite sum exists.
Apply the formula:
S∞ = a/(1 − r)
S∞ = 8/(1 − 1/2)
S∞ = 8/(1/2)
S∞ = 16
Final Answer: 16
What If |r| ≥ 1?
For an ordinary real infinite GP, the series does not have a finite sum when |r| is greater than or equal to 1, apart from the trivial all-zero sequence where a = 0. For the standard non-zero GP questions in JEE Main, check |r| < 1 before applying the infinite-sum formula.
JEE Main Trick: Infinite GP ka formula apply karne se pehle common ratio ka absolute value check karo. The condition |r| < 1 is essential.
5. Important GP Properties and Shortcuts
1. Three Consecutive Terms in GP
Three consecutive terms of a GP can be represented as:
a/r, a, ar
Here, a is the middle term and r is the common ratio.
2. Square of the Middle Term
For three consecutive terms a/r, a, ar:
a² = (a/r)(ar)
In general, if a, b, c are three consecutive terms of a GP, then:
b² = ac
This property is useful when the question gives three consecutive terms but does not provide the common ratio directly.
3. Geometric Mean
For two positive numbers a and b, the geometric mean is:
GM = √ab
If a, G, b are positive numbers in GP, then G is the geometric mean of a and b:
G² = ab
G = √ab
For positive a and b, the geometric mean is the positive square root. In a more general GP with negative terms, use the GP condition and sign information given in the question.
4. Common Ratio
The common ratio is calculated by dividing a term by the previous term:
r = Second term / First term
More generally:
r = aₙ₊₁/aₙ
provided the denominator term is non-zero.
5. Negative Common Ratio
The common ratio of a GP can be negative. For example:
2, −4, 8, −16, ...
Calculate the ratio:
r = −4/2 = −2
Each term alternates in sign because the common ratio is negative.
Exam Trick: Do not assume that the common ratio must be positive. Check the actual ratio between consecutive terms.
6. JEE Main Solved Question: Sum of the First 6 Terms
Question
Find the sum of the first 6 terms of the GP:
3, 6, 12, 24, ...
Step 1: Identify the Given Values
- First term: a = 3
- Common ratio: r = 6/3 = 2
- Number of terms: n = 6
Step 2: Choose the Sum Formula
Since r ≠ 1, use:
Sₙ = a(rⁿ − 1)/(r − 1)
Step 3: Substitute the Values
S₆ = 3(2⁶ − 1)/(2 − 1)
S₆ = 3(64 − 1)/1
S₆ = 3 × 63
S₆ = 189
Final Answer: 189
JEE Main Learning
- Identify a, r and n before applying the formula.
- Use the finite-sum formula because the question asks for the first 6 terms.
- Do not use the infinite-sum formula for a finite number of terms.
7. Common Mistakes and JEE Main Exam Tricks
Mistake 1: Using n Instead of n − 1 in the Nth-Term Formula
Incorrect:
aₙ = arⁿ
Correct:
aₙ = arⁿ⁻¹
The exponent is one less than the position of the term.
Mistake 2: Confusing the Nth Term with the Sum
If the question asks for one particular term, use aₙ = arⁿ⁻¹. If it asks for the sum of the first n terms, use the finite-sum formula.
Mistake 3: Using the Wrong Formula When r = 1
When every term is the same, use Sₙ = na. Do not substitute r = 1 into the general formula with denominator r − 1.
Mistake 4: Forgetting the Infinite-Sum Condition
The formula S∞ = a/(1 − r) is valid only when |r| < 1 for the standard infinite GP.
Mistake 5: Assuming the Common Ratio Must Be Positive
A GP can have a negative common ratio, as in 2, −4, 8, −16, ...
Mistake 6: Confusing Finite and Infinite Series
If a question gives a fixed number of terms, use the finite-sum formula. Use the infinite-sum formula only when the series continues indefinitely and satisfies the required condition.
Mistake 7: Not Identifying Given Values First
Before solving, write down a, r and n wherever applicable. This simple step helps you select the right formula and avoid substitution errors.
Important JEE Main Shortcuts
- GP condition: Consecutive terms have a constant ratio.
- Nth term: aₙ = arⁿ⁻¹.
- Finite sum for r ≠ 1: Sₙ = a(rⁿ − 1)/(r − 1).
- Finite sum for r = 1: Sₙ = na.
- Infinite sum: S∞ = a/(1 − r), when |r| < 1.
- Three consecutive GP terms: a/r, a, ar.
- For three consecutive terms a, b, c: b² = ac.
- For positive a and b: GM = √ab.
8. Quick Revision Formula Sheet
| Concept | Formula or Rule |
|---|---|
| General form of GP | a, ar, ar², ar³, ... |
| Common ratio | r = Second term / First term |
| Nth term | aₙ = arⁿ⁻¹ |
| Sum of first n terms, r ≠ 1 | Sₙ = a(rⁿ − 1)/(r − 1) |
| Equivalent finite-sum formula | Sₙ = a(1 − rⁿ)/(1 − r) |
| Sum when r = 1 | Sₙ = na |
| Sum of an infinite GP | S∞ = a/(1 − r), when |r| < 1 |
| Three consecutive GP terms | a/r, a, ar |
| Three consecutive terms a, b, c | b² = ac |
| Geometric mean of positive a and b | GM = √ab |
9. One-Minute Revision
- GP: Consecutive terms have a constant common ratio.
- General form: a, ar, ar², ...
- Common ratio: r = a₂/a₁, when a₁ ≠ 0.
- Nth term: aₙ = arⁿ⁻¹.
- Finite sum: Sₙ = a(rⁿ − 1)/(r − 1), when r ≠ 1.
- If r = 1: Sₙ = na.
- Infinite sum: S∞ = a/(1 − r), when |r| < 1.
- Three consecutive terms: a/r, a, ar.
- GP condition: For three consecutive terms a, b, c, b² = ac.
- Geometric mean: For positive a and b, GM = √ab.
10. Practice Questions for JEE Main
Question 1: Nth Term
Find the 5th term of the GP 2, 6, 18, 54, ...
- 108
- 162
- 324
- 486
Answer: (b) 162
Solution: a = 2, r = 3 and n = 5. Therefore, a₅ = 2 × 3⁴ = 2 × 81 = 162.
Question 2: Finite Sum
Find the sum of the first 4 terms of the GP 1, 2, 4, 8, ...
- 12
- 15
- 16
- 18
Answer: (b) 15
Solution: 1 + 2 + 4 + 8 = 15.
Question 3: Infinite GP
Find the sum of 12 + 6 + 3 + 3/2 + ...
- 18
- 20
- 24
- 30
Answer: (c) 24
Solution: a = 12 and r = 1/2. Since |r| < 1, S∞ = 12/(1 − 1/2) = 24.
Question 4: Common Ratio
What is the common ratio of the GP 3, −6, 12, −24, ...?
- 2
- −2
- 3
- −3
Answer: (b) −2
Solution: r = −6/3 = −2.
Question 5: Three Consecutive GP Terms
If 4, x, 16 are three consecutive positive terms of a GP, find x.
- 4
- 6
- 8
- 12
Answer: (c) 8
Solution: x² = 4 × 16 = 64. Since x is positive, x = 8.
11. Sequence and Series Revision PDF
Embed your GP notes or revision PDF below. Replace the placeholder with the direct URL of your uploaded PDF when it is available.
12. Frequently Asked Questions (FAQs)
Q1. What is a Geometric Progression?
A Geometric Progression is a sequence in which the ratio of consecutive terms remains constant. This constant is called the common ratio.
Q2. What is the formula for the nth term of a GP?
The nth-term formula is aₙ = arⁿ⁻¹, where a is the first term and r is the common ratio.
Q3. What is the formula for the sum of the first n terms?
For r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1). When r = 1, use Sₙ = na.
Q4. What is the formula for the sum of an infinite GP?
The formula is S∞ = a/(1 − r), provided |r| < 1.
Q5. Can the common ratio of a GP be negative?
Yes. For example, 2, −4, 8, −16, ... is a GP with common ratio −2.
Q6. What is the condition for three numbers to be consecutive terms of a GP?
For three consecutive terms a, b, c, the condition is b² = ac.
Q7. What is the geometric mean of two positive numbers?
The geometric mean of two positive numbers a and b is GM = √ab.
Q8. What is the difference between the nth term and the sum of n terms?
The nth-term formula gives one particular term, while the sum formula gives the total of the first n terms.
Q9. Why must |r| be less than 1 for an infinite GP to have a finite sum?
When |r| < 1, successive terms approach zero, allowing the partial sums to approach a finite limit. For a standard non-zero infinite GP with |r| ≥ 1, a finite sum does not exist.
Q10. Which GP formulas should I memorise for JEE Main?
Focus on the nth-term formula, finite-sum formula, infinite-sum formula and its condition, common ratio, and the relation b² = ac for three consecutive terms.
Final Thoughts
Geometric Progression is a formula-based topic that becomes easier with consistent practice. First identify the first term and common ratio, then determine whether the question asks for a particular term, a finite sum, or an infinite sum.
For JEE Main 2027, practise questions involving nth terms, finite sums, infinite GP, negative common ratios and the geometric mean. Always check the conditions before applying a formula.
Final JEE Main Tip: Nth term ke liye aₙ = arⁿ⁻¹, finite sum ke liye Sₙ, aur infinite sum ke liye S∞ use karo. Formula choose karne se pehle question exactly kya pooch raha hai, ye identify karna sabse important step hai.