JEE Prep Guide

JEE Prep Guide

Path to IIT

📑 Quick Navigation - Sequences and Series

Basics AP GP HP AM-GM-HM AGP Special Series Infinite Series Applications

Sequences and Series JEE Main & Advanced 2025-26

Master the highest weightage chapter with complete notes on AP, GP, HP, AGP, AM-GM-HM, Infinite Series & Special Summations. Includes 250+ solved problems and all JEE shortcuts.

📚 9 Complete Sections
✍️ 250+ Solved Examples
🎯 All Formulas
⭐⭐⭐⭐⭐ 4.9/5
428+ students rated this helpful
250+
Solved Examples
5 hrs
Study Time
Free
Always
1

Sequence and Series Fundamentals

Sequences and Series form the foundation of many mathematical concepts and have extensive applications in calculus, physics, and engineering. This is one of the highest weightage topics in JEE.

1.1 What is a Sequence?

Definition of Sequence

A sequence is an ordered list of numbers arranged according to a definite rule. Each number in the sequence is called a term.

General Form:

a₁, a₂, a₃, a₄, ..., aₙ, ...

where:

  • a₁ = first term
  • a₂ = second term
  • aₙ = nth term (general term)
  • n = position/index of term
✅ Examples of Sequences

1. Even natural numbers:

2, 4, 6, 8, 10, ...

Rule: aₙ = 2n

2. Squares:

1, 4, 9, 16, 25, ...

Rule: aₙ = n²

3. Powers of 2:

2, 4, 8, 16, 32, ...

Rule: aₙ = 2ⁿ

4. Fibonacci:

1, 1, 2, 3, 5, 8, ...

Rule: aₙ = aₙ₋₁ + aₙ₋₂

Types of Sequences

Finite Sequence

Has a last term

1, 2, 3, 4, 5

Infinite Sequence

Continues forever

1, 2, 3, 4, ...

Constant Sequence

All terms equal

5, 5, 5, 5, ...

1.2 What is a Series?

Definition of Series

A series is the sum of terms of a sequence. If a₁, a₂, a₃, ... is a sequence, then the corresponding series is:

\[a_1 + a_2 + a_3 + ... + a_n + ... = \sum_{i=1}^{n} a_i\]

Key Notation:

  • Σ (Sigma) - Summation symbol
  • Sₙ - Sum of first n terms
  • S∞ - Sum to infinity (if exists)
Sequence vs Series
Aspect Sequence Series
Definition Ordered list Sum of terms
Symbol aₙ Sₙ or Σ
Example 1,2,3,4 1+2+3+4=10
Sigma (Σ) Notation
\[\sum_{i=1}^{n} a_i = a_1 + a_2 + ... + a_n\]

Parts of Sigma Notation:

  • i = 1 (lower limit)
  • n (upper limit)
  • aᵢ (general term)

📝 Solved Example 1

Question: Find the sum: \(\sum_{k=1}^{5} (2k + 1)\)

Solution:

Step 1: Expand the summation

When k = 1: 2(1) + 1 = 3

When k = 2: 2(2) + 1 = 5

When k = 3: 2(3) + 1 = 7

When k = 4: 2(4) + 1 = 9

When k = 5: 2(5) + 1 = 11

Step 2: Add all terms

Sum = 3 + 5 + 7 + 9 + 11

Sum = 35

Answer: 35

1.3 Important Sigma Properties

Properties of Summation (Must Know for JEE)

1. Sum of Constant

\[\sum_{i=1}^{n} c = nc\]

2. Constant Multiple

\[\sum_{i=1}^{n} ca_i = c\sum_{i=1}^{n} a_i\]

3. Sum Property

\[\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i\]

4. Difference Property

\[\sum_{i=1}^{n} (a_i - b_i) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i\]

💡 Important Points to Remember

  • Sequence: Focus on individual terms and their pattern
  • Series: Focus on sum of terms
  • Notation: Use Σ for series, subscript notation for sequences
  • nth term: Finding general term is crucial for solving problems
  • Index: Can start from any value, not necessarily 1
2

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. This is one of the most important and frequently tested topics in JEE.

2.1 Definition and General Form

Arithmetic Progression (AP)

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

General Form:

a, a+d, a+2d, a+3d, ..., a+(n-1)d

a = First term

d = Common difference

n = Number of terms

l = Last term = a+(n-1)d

Common Difference Formula:

\[d = a_{n+1} - a_n = a_2 - a_1\]

2.2 nth Term Formula

⭐ MOST IMPORTANT - nth Term of AP

\[T_n = a + (n-1)d\]

Variations:

• Last term: l = a + (n-1)d

• Middle term (odd n): T(n+1)/2

• Middle terms (even n): Tn/2 and T(n/2)+1

📝 Solved Example 2

Question: Find the 20th term of the AP: 5, 8, 11, 14, ...

Solution:

Given AP: 5, 8, 11, 14, ...

Step 1: Identify first term and common difference

First term (a) = 5

Common difference (d) = 8 - 5 = 3

We need: n = 20

Step 2: Apply nth term formula

Tn = a + (n-1)d

T20 = 5 + (20-1)×3

T20 = 5 + 19×3

T20 = 5 + 57

T20 = 62

Answer: 20th term = 62

2.3 Sum of n Terms in AP

⭐ Sum Formulas (Extremely Important)

Formula 1:

\[S_n = \frac{n}{2}[2a + (n-1)d]\]

Formula 2:

\[S_n = \frac{n}{2}(a + l)\]

When to use which formula:

  • Formula 1: When last term is not given (use a, d, n)
  • Formula 2: When last term is given (use a, l, n)
  • Both give same result - choose based on given data

📝 Solved Example 3 (JEE Main Pattern)

Question: Find the sum of first 50 terms of the AP: 2, 5, 8, 11, ...

Solution:

Given: a = 2, d = 3, n = 50

Method 1: Using first formula

Sn = n/2[2a + (n-1)d]

S50 = 50/2[2(2) + (50-1)×3]

S50 = 25[4 + 49×3]

S50 = 25[4 + 147]

S50 = 25 × 151

S50 = 3775

Method 2: Using second formula

First find last term: l = a + (n-1)d

l = 2 + 49×3 = 2 + 147 = 149

Sn = n/2(a + l)

S50 = 50/2(2 + 149)

S50 = 25 × 151 = 3775

Answer: Sum = 3775

2.4 Important Properties of AP

Properties (Very Important for JEE)

1. Selection of Terms in AP

Number of Terms Selection Sum
3 terms a-d, a, a+d 3a
4 terms a-3d, a-d, a+d, a+3d 4a
5 terms a-2d, a-d, a, a+d, a+2d 5a

2. Three consecutive terms a, b, c are in AP if:

\[2b = a + c \text{ or } b - a = c - b\]

3. If each term is increased/decreased by same constant:

Resulting sequence is also in AP with same common difference

4. If each term is multiplied/divided by same non-zero constant:

Resulting sequence is in AP with common difference multiplied/divided by that constant

5. Sum of terms equidistant from beginning and end:

\[T_k + T_{n-k+1} = a + l = \text{constant}\]

💡 JEE Shortcuts for AP

Quick Formulas:

  • Tn = Sn - Sn-1
  • Middle term (odd): T(n+1)/2
  • Sum of n natural numbers: n(n+1)/2
  • Sum of first n odd numbers: n²
  • Sum of first n even numbers: n(n+1)

Common Patterns:

  • If d > 0: AP is increasing
  • If d < 0: AP is decreasing
  • If d = 0: Constant sequence
  • AP with a=1, d=1: Natural numbers
  • AP with a=2, d=2: Even numbers
3

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant. This is crucial for exponential growth/decay problems in JEE.

3.1 Definition and General Form

Geometric Progression (GP)

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

General Form:

a, ar, ar², ar³, ..., arⁿ⁻¹

a = First term (a ≠ 0)

r = Common ratio (r ≠ 0)

n = Number of terms

l = Last term = arⁿ⁻¹

Common Ratio Formula:

\[r = \frac{a_{n+1}}{a_n} = \frac{a_2}{a_1}\]

3.2 nth Term Formula

⭐ nth Term of GP

\[T_n = ar^{n-1}\]

Important Cases:

• If r > 1 and a > 0: GP is increasing

• If 0 < r < 1 and a > 0: GP is decreasing

• If r < 0: GP has alternating signs

• If r = 1: All terms equal (constant sequence)

• If r = -1: Terms alternate between a and -a

📝 Solved Example 4

Question: Find the 8th term of the GP: 3, 6, 12, 24, ...

Given GP: 3, 6, 12, 24, ...

Step 1: Find a and r

First term (a) = 3

Common ratio (r) = 6/3 = 2

We need: n = 8

Step 2: Apply nth term formula

Tn = arn-1

T8 = 3 × 28-1

T8 = 3 × 27

T8 = 3 × 128

T8 = 384

Answer: 8th term = 384

3.3 Sum of n Terms in GP

⭐ Sum Formulas (Most Important)

When r ≠ 1:

\[S_n = \frac{a(r^n - 1)}{r - 1} = \frac{a(1 - r^n)}{1 - r}\]

When r = 1:

\[S_n = na\]

When to use which form:

  • Use a(rⁿ-1)/(r-1): When r > 1
  • Use a(1-rⁿ)/(1-r): When 0 < r < 1
  • Both are algebraically equivalent, choose for easier calculation

3.4 Sum to Infinity (S∞)

⭐ Infinite GP Sum (Very Important for JEE Advanced)

Convergence Condition:

An infinite GP converges (has finite sum) if and only if |r| < 1

\[S_\infty = \frac{a}{1-r} \quad \text{when } |r| < 1\]

✅ Converges (|r| < 1)

Examples:

1 + 1/2 + 1/4 + 1/8 + ...

Sum = 2

❌ Diverges (|r| ≥ 1)

Examples:

1 + 2 + 4 + 8 + ...

Sum = ∞ (infinite)

📝 Solved Example 5 (JEE Advanced Pattern)

Question: Find the sum to infinity: 1 + 1/3 + 1/9 + 1/27 + ...

Given: 1, 1/3, 1/9, 1/27, ...

Step 1: Identify a and r

a = 1

r = (1/3)/1 = 1/3

Step 2: Check convergence condition

|r| = |1/3| = 1/3 < 1 ✓

Since |r| < 1, infinite sum exists

Step 3: Apply infinite sum formula

S∞ = a/(1-r)

S∞ = 1/(1 - 1/3)

S∞ = 1/(2/3)

S∞ = 3/2

Answer: S∞ = 3/2

3.5 Properties of GP

Important GP Properties

1. Selection of Terms in GP

Number of Terms Selection Product
3 terms a/r, a, ar a³
4 terms a/r³, a/r, ar, ar³ a⁴
5 terms a/r², a/r, a, ar, ar² a⁵

2. Three numbers a, b, c in GP if:

\[b^2 = ac \quad \text{or} \quad \frac{b}{a} = \frac{c}{b}\]

3. If each term is multiplied/divided by same constant k:

New sequence is also GP with same ratio r, but first term becomes ak or a/k

4. Product of terms equidistant from ends:

\[T_k \times T_{n-k+1} = a \times l = \text{constant}\]

5. Reciprocals of GP terms:

Form a GP with common ratio 1/r

⚠️ Common Mistakes in GP

  • Forgetting r ≠ 1 condition: When r = 1, use Sₙ = na, not the GP formula
  • |r| < 1 for infinite sum: Must check convergence before finding S∞
  • Sign of r: Negative r gives alternating series
  • Zero terms not allowed: Neither a nor r can be zero in GP
  • Power confusion: nth term is arn-1, not arn
4

Harmonic Progression (HP)

Harmonic Progression Definition

A sequence is in HP if the reciprocals of its terms form an AP.

If HP: a, b, c, d, ...

Then AP: 1/a, 1/b, 1/c, 1/d, ...

\[\text{nth term of HP: } T_n = \frac{1}{a + (n-1)d}\]

where a and d are first term and common difference of corresponding AP

🎯 Important HP Properties

  • No direct formula for sum of HP (convert to AP)
  • Three terms a, b, c in HP if: 2/b = 1/a + 1/c
  • HP is always between corresponding AP and GP terms
5

AM-GM-HM Relations

⭐ Means and Inequality

Arithmetic Mean (AM)

\[AM = \frac{a + b}{2}\]

Geometric Mean (GM)

\[GM = \sqrt{ab}\]

Harmonic Mean (HM)

\[HM = \frac{2ab}{a + b}\]

AM-GM-HM Inequality

\[AM \geq GM \geq HM\]

Equality holds when a = b

Important Relation:

\[GM^2 = AM \times HM\]
6

Arithmetic-Geometric Progression (AGP)

AGP Definition and Sum

AGP is formed by multiplying corresponding terms of an AP and GP.

General Form:

a, (a+d)r, (a+2d)r², (a+3d)r³, ...

Sum of n terms:

\[S_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{a+(n-1)d}{1-r}r^n\]

Sum to infinity (|r| < 1):

\[S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}\]
7

Special Series Formulas

⭐ Must Know Special Series

1. Sum of first n natural numbers

\[\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\]

2. Sum of squares

\[\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\]

3. Sum of cubes

\[\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2\]

4. Sum of first n odd numbers

\[\sum_{k=1}^{n} (2k-1) = n^2\]

5. Sum of first n even numbers

\[\sum_{k=1}^{n} 2k = n(n+1)\]

6. Sum relation

\[\sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2\]

📝 Previous Year Questions Analysis

JEE Main (Last 5 Years)

  • ✓ AP Sum & nth term: 30%
  • ✓ GP & Infinite Series: 35%
  • ✓ AM-GM-HM Applications: 20%
  • ✓ Special Series: 15%

JEE Advanced (Last 5 Years)

  • ✓ AGP & Complex Series: 30%
  • ✓ Infinite GP Convergence: 25%
  • ✓ Mixed AP-GP Problems: 25%
  • ✓ Special Series Summation: 20%

Top 15 Most Repeated Question Types

  1. Finding nth term of AP/GP given conditions
  2. Sum of first n terms problems
  3. Infinite GP sum and convergence
  4. AM-GM-HM inequality applications
  5. Finding missing terms in progression
  6. Word problems on AP/GP
  7. AGP sum to n terms
  8. Sum of special series (squares, cubes)
  9. Geometric mean insertion
  10. Arithmetic mean insertion
  11. Sum of infinite series
  12. Proving sequence is AP/GP
  13. Mixed progression problems
  14. Telescoping series
  15. Sigma notation problems

Weightage Analysis

JEE Main: 16-20 marks (4-5 questions)
JEE Advanced: 12-15 marks (2-3 questions)
Difficulty Level: Easy to Very Hard
Time Required for Prep: 5-6 hours

🎯 Practice Problem Set

Level 1: Basic (Foundation)

  1. Find 15th term of AP: 3, 7, 11, 15, ...
  2. Find sum of first 20 terms of AP: 5, 10, 15, ...
  3. Find 10th term of GP: 2, 6, 18, ...
  4. Check if 2, 8, 32 are in GP
  5. Find AM, GM, HM of 4 and 16
  6. Sum: 1 + 2 + 3 + ... + 100 = ?
  7. Find common difference if 5th term is 17 and 8th term is 26
  8. Find sum to infinity: 1 + 1/2 + 1/4 + 1/8 + ...

Level 2: Intermediate (JEE Main)

  1. Sum of n terms of AP is 3n² + 5n. Find nth term.
  2. Three numbers in GP have sum 21 and product 216. Find them.
  3. Insert 5 AM's between 3 and 21
  4. Find sum: 1² + 2² + 3² + ... + 50²
  5. If a, b, c are in HP, prove b = 2ac/(a+c)
  6. Sum of n terms: 1 + 3 + 5 + 7 + ... = ?
  7. Find GP if 2nd term = 6 and 5th term = 162
  8. Prove AM ≥ GM for two positive numbers

Level 3: Advanced (JEE Advanced)

  1. Find sum to infinity: 1 + 2x + 3x² + 4x³ + ... (|x| < 1)
  2. Sum of n terms of AGP: 1·2 + 2·4 + 3·8 + ...
  3. If a, b, c, d are in GP, prove (a²+b²+c²)(b²+c²+d²) = (ab+bc+cd)²
  4. Find sum: 1·2 + 2·3 + 3·4 + ... + n(n+1)
  5. If sum of n terms of AP is 3n²+n and sum to m terms is 3m²+m, find sum to (m+n) terms
  6. Prove: 1³+2³+3³+...+n³ = (1+2+3+...+n)²
  7. Sum: 1/1·2 + 1/2·3 + 1/3·4 + ... to n terms
  8. If a, b, c are in AP and a², b², c² are in HP, prove a, b, c are equal
Download Complete Solutions PDF

Sequences and Series - Complete Guide for JEE 2025-26

Why Sequences and Series is CRUCIAL for JEE?

Sequences and Series carries the highest weightage (6-7%) among all algebra topics:

  • Direct Questions: 4-5 questions every year
  • Binomial Theorem: Uses GP and special series
  • Calculus: Foundation for limits and series
  • Probability: GP in infinite events
  • Mathematical Induction: Proving series formulas

Most Important Formulas

AP nth term:

Tₙ = a + (n-1)d

AP Sum:

Sₙ = n/2[2a + (n-1)d]

GP nth term:

Tₙ = arⁿ⁻¹

GP Sum:

Sₙ = a(rⁿ-1)/(r-1)

Infinite GP:

S∞ = a/(1-r), |r|<1

AM-GM-HM:

AM ≥ GM ≥ HM