Sequences and Series JEE Notes 2026 | AP, GP, HP, AM-GM-HM, AGP, Telescopic Series PYQs

Q: What is a Telescopic Series?

A telescopic series is one where most terms cancel out when expanded. It is usually solved using partial fractions, leaving only the first and last terms, making the sum easy to calculate.

1

Introduction to Sequences & Series

A sequence is an ordered list of numbers arranged according to a specific rule. A series is the sum of the terms of a sequence. Sequences and Series is a foundational algebra chapter for JEE, typically yielding 1-2 questions per session.

1.1 Sequence

A sequence is a function whose domain is the set of natural numbers (or a subset of it) and whose range is a subset of real numbers. It is denoted by \(a_1, a_2, a_3, \dots, a_n, \dots\) or simply \(\{a_n\}\).

Finite Sequence: Has a limited number of terms (e.g., 2, 4, 6, 8).
Infinite Sequence: Continues indefinitely (e.g., 2, 4, 6, 8, ...).

1.2 Series

If \(a_1, a_2, a_3, \dots, a_n\) is a sequence, then the expression \(a_1 + a_2 + a_3 + \dots + a_n\) is called a series.

S_n = \sum_{k=1}^{n} a_k

1.3 Progression

A sequence is called a progression if its terms follow a specific pattern (like adding a constant or multiplying by a constant). Common progressions are AP, GP, and HP.

PYQ — JEE Main 2021 (Mar 17 Shift 1)

Question: If the sum of the first 20 terms of the series \( \log_{10}(7^{1/2}) + \log_{10}(7^{1/3}) + \log_{10}(7^{1/4}) + \dots \) is equal to \( \log_{10}(7^k) \), then find the value of \(k\).

Solution:

The series can be rewritten using logarithm properties: \(\frac{1}{2}\log 7 + \frac{1}{3}\log 7 + \frac{1}{4}\log 7 + \dots\)

Factor out \(\log 7\): \(\log 7 \left( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{21} \right)\).

The sum inside the bracket is the sum of 20 terms starting from \(\frac{1}{2}\).

k = \sum_{r=2}^{21} \frac{1}{r} = H_{21} - 1

So, \(k = \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{21}\).

Answer: \(k = \sum_{n=2}^{21} \frac{1}{n}\)

2

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. This constant is called the common difference (d).

2.1 Standard Form & General Term

Most Repeated PYQ — General Term of AP

Standard form: \(a, a+d, a+2d, \dots\)

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

Where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

2.2 Sum of n Terms of AP

Most Repeated PYQ — Sum of AP

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

where \(l\) is the last term.

2.3 Properties of AP

If terms of an AP are added, subtracted, multiplied, or divided by the same constant, the resulting sequence is also an AP.
If \(T_n = an^2 + bn + c\), then the sequence is an AP with common difference \(2a\). (Sum \(S_n\) is a cubic in \(n\)).
Sum of equidistant terms from beginning and end is constant: \(T_k + T_{n-k+1} = T_1 + T_n\).
If \(S_n\) is the sum of \(n\) terms, then \(T_n = S_n - S_{n-1}\).

2.4 Arithmetic Mean (AM)

Important — Arithmetic Mean

If \(a, A, b\) are in AP, then \(A\) is the Arithmetic Mean of \(a\) and \(b\).

A = \frac{a+b}{2}

For \(n\) numbers, the AM is \(\frac{a_1 + a_2 + \dots + a_n}{n}\). To insert \(n\) AMs between \(a\) and \(b\), the common difference is \(d = \frac{b-a}{n+1}\).

PYQ — JEE Main 2023 (Apr 10 Shift 2)

Question: Let \(a_1, a_2, \dots, a_{100}\) be an AP with \(a_1 = 3\) and sum of first 100 terms \(S_{100} = 505\). Find the common difference.

Solution:

Given \(a = 3\), \(n = 100\), \(S_{100} = 505\).

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

505 = \frac{100}{2} [2(3) + 99d]

\[505 = 50 [6 + 99d]\]

10.1 = 6 + 99d \Rightarrow 4.1 = 99d

d = \frac{4.1}{99} = \frac{41}{990}

Answer: \(d = \frac{41}{990}\)

3

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence where the ratio of consecutive terms is constant. This constant is the common ratio (r).

3.1 General Term of GP

Most Repeated PYQ — General Term of GP

Standard form: \(a, ar, ar^2, \dots\)

T_n = ar^{n-1}

3.2 Sum of n Terms of GP

Most Repeated PYQ — Sum of Finite GP

S_n = \frac{a(r^n - 1)}{r - 1} \quad (\text{for } r > 1)\] \[S_n = \frac{a(1 - r^n)}{1 - r} \quad (\text{for } r < 1)

If \(r = 1\), \(S_n = na\).

3.3 Sum of Infinite GP

Most Repeated PYQ — Infinite GP

For an infinite GP \(a, ar, ar^2, \dots\), the sum converges if \(|r| < 1\):

S_\infty = \frac{a}{1 - r}

If \(|r| \ge 1\), the sum diverges to infinity.

3.4 Geometric Mean (GM)

Important — Geometric Mean

If \(a, G, b\) are in GP, then \(G\) is the Geometric Mean of \(a\) and \(b\).

G = \sqrt{ab}

To insert \(n\) GMs between \(a\) and \(b\), the common ratio is \(r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}\).

3.5 Properties of GP

If terms of a GP are multiplied or divided by the same non-zero constant, the resulting sequence is also a GP.
The reciprocals of a GP also form a GP with common ratio \(1/r\).
If each term of a GP is raised to the same power \(k\), the resulting sequence is a GP with common ratio \(r^k\).
Product of equidistant terms from beginning and end is constant: \(T_k \cdot T_{n-k+1} = T_1 \cdot T_n\).

PYQ — JEE Main 2024 (Jan 30 Shift 1)

Question: If the sum of an infinite GP is 3 and the sum of the squares of its terms is also 3, find the first term and common ratio.

Solution:

Let the GP be \(a, ar, ar^2, \dots\)

Given \(S_\infty = \frac{a}{1-r} = 3 \Rightarrow a = 3(1-r)\).

Squares of terms: \(a^2, a^2r^2, a^2r^4, \dots\). This is a GP with first term \(a^2\) and ratio \(r^2\).

\frac{a^2}{1-r^2} = 3

Substitute \(a = 3(1-r)\):

\frac{9(1-r)^2}{1-r^2} = 3 \Rightarrow \frac{9(1-r)^2}{(1-r)(1+r)} = 3

\frac{3(1-r)}{1+r} = 1 \Rightarrow 3 - 3r = 1 + r \Rightarrow 2 = 4r \Rightarrow r = 1/2

Then \(a = 3(1 - 1/2) = 3/2\).

Answer: \(a = 3/2, r = 1/2\)

4

Harmonic Progression (HP)

A sequence is said to be in Harmonic Progression (HP) if the reciprocals of its terms form an Arithmetic Progression (AP).

4.1 General Term of HP

Most Repeated PYQ — General Term of HP

If \(a, a+d, a+2d, \dots\) is an AP, then \(\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \dots\) is an HP.

T_n = \frac{1}{a + (n-1)d}

Note: A term of an HP can never be zero.

4.2 Harmonic Mean (HM)

Important — Harmonic Mean

If \(a, H, b\) are in HP, then \(H\) is the Harmonic Mean of \(a\) and \(b\).

H = \frac{2ab}{a+b}

To insert \(n\) HMs between \(a\) and \(b\), the corresponding AP has common difference \(d = \frac{b-a}{(n+1)ab}\).

PYQ — JEE Main 2020 (Sep 5 Shift 1)

Question: If \(H\) is the harmonic mean between \(a\) and \(b\), then find the value of \(\frac{H}{a} + \frac{H}{b}\).

Solution:

Given \(H = \frac{2ab}{a+b}\).

Calculate \(\frac{H}{a} + \frac{H}{b}\):

H\left(\frac{1}{a} + \frac{1}{b}\right) = \frac{2ab}{a+b} \cdot \left(\frac{a+b}{ab}\right) = 2

Answer: 2

5

Relation between AM, GM, and HM

For any two positive numbers, the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) satisfy specific inequalities and relations.

5.1 Inequality AM ≥ GM ≥ HM

Most Repeated PYQ — Inequality

For any two positive real numbers \(a\) and \(b\):

AM \ge GM \ge HM

Equality holds if and only if \(a = b\).

5.2 Relation AM, GM, HM

Most Repeated PYQ — AM, GM, HM Formula

For two positive numbers, the square of the Geometric Mean is equal to the product of the Arithmetic Mean and the Harmonic Mean:

GM^2 = AM \times HM

This means \(G^2 = A \cdot H\), or equivalently, \(A, G, H\) are in GP.

5.3 General Formulas for Means

AM of n numbers: \(\frac{a_1 + a_2 + \dots + a_n}{n}\)
GM of n numbers: \((a_1 \cdot a_2 \dots a_n)^{1/n}\)
HM of n numbers: \(\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}}\)

PYQ — JEE Main 2022 (June 28 Shift 1)

Question: If the AM of two positive numbers is 3 and their GM is \( \sqrt{5} \), find their HM.

Solution:

Given \(AM = A = 3\), \(GM = G = \sqrt{5}\).

Using \(G^2 = A \cdot H\):

(\sqrt{5})^2 = 3 \cdot H \Rightarrow 5 = 3H

H = \frac{5}{3}

Answer: \(5/3\)

6

Arithmetico-Geometric Progression (AGP)

An Arithmetico-Geometric Progression (AGP) is a sequence where each term is formed by multiplying the corresponding terms of an AP and a GP.

6.1 Standard Form

Most Repeated PYQ — AGP Sequence

If AP is \(a, a+d, a+2d, \dots\) and GP is \(1, r, r^2, \dots\), the AGP is:

a, (a+d)r, (a+2d)r^2, \dots, (a+(n-1)d)r^{n-1}

6.2 Sum of n Terms of AGP

Most Repeated PYQ — Sum of AGP

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

6.3 Sum to Infinity of AGP

Most Repeated PYQ — Infinite AGP

If \(|r| < 1\), the sum to infinity converges to:

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

PYQ — JEE Main 2023 (Jan 31 Shift 2)

Question: Find the sum to infinity of the series \(1 + \frac{2}{3} + \frac{3}{3^2} + \frac{4}{3^3} + \dots\)

Solution:

This is an infinite AGP with \(a = 1\), \(d = 1\), \(r = 1/3\).

Using \(S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}\):

S_\infty = \frac{1}{1-1/3} + \frac{1 \cdot (1/3)}{(1-1/3)^2}

S_\infty = \frac{1}{2/3} + \frac{1/3}{4/9} = \frac{3}{2} + \frac{1}{3} \times \frac{9}{4} = \frac{3}{2} + \frac{3}{4}

S_\infty = \frac{6}{4} + \frac{3}{4} = \frac{9}{4}

Answer: \(9/4\)

7

Sum of Special Series

The sums of the first \(n\) natural numbers, their squares, and their cubes are standard formulas frequently used in JEE.

7.1 Sum of First n Natural Numbers

Most Repeated PYQ — Sum of Natural Numbers

\sum_{k=1}^{n} k = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}

7.2 Sum of Squares of First n Natural Numbers

Most Repeated PYQ — Sum of Squares

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

7.3 Sum of Cubes of First n Natural Numbers

Most Repeated PYQ — Sum of Cubes

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

PYQ — JEE Main 2024 (Apr 5 Shift 2)

Question: Find the sum of the series \(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots\) to 10 terms.

Solution:

General term \(T_n = n(n+1) = n^2 + n\).

Sum \(S_n = \sum (n^2 + n) = \sum n^2 + \sum n\).

S_{10} = \frac{10 \cdot 11 \cdot 21}{6} + \frac{10 \cdot 11}{2}

S_{10} = 385 + 55 = 440

Answer: 440

8

Telescopic Series

A Telescopic Series is one where terms cancel out when expanded, usually leaving only the first and last terms.

8.1 Partial Fractions Method

Most Repeated PYQ — Telescopic Sum

To find the sum of series like \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n+1)} \), we use partial fractions:

\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}

So the sum becomes:

S_n = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \dots + \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1 - \frac{1}{n+1} = \frac{n}{n+1}

8.2 Factorial Series

For terms involving factorials, use the property:

\frac{n}{(n+1)!} = \frac{(n+1)-1}{(n+1)!} = \frac{1}{n!} - \frac{1}{(n+1)!}

PYQ — JEE Main 2024 (Apr 6 Shift 1)

Question: Find the sum to infinity of the series \( \frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \dots \)

Solution:

General term \(T_n = \frac{1}{(2n-1)(2n+1)} = \frac{1}{2} \left[ \frac{1}{2n-1} - \frac{1}{2n+1} \right]\).

Sum \(S_\infty = \frac{1}{2} \left[ \left(1 - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \dots \right]\).

All terms cancel except 1.

S_\infty = \frac{1}{2} \cdot 1 = \frac{1}{2}

Answer: \(1/2\)

9

Method of Differences

If a series does not fall directly into AP, GP, or standard series, the Method of Differences is used by finding the difference between consecutive terms.

9.1 Sum using Differences

Most Repeated PYQ — Method of Differences

Let \(S_n = T_1 + T_2 + \dots + T_n\). Find \(T_n - T_{n-1}\). If this difference forms an AP or GP, we can write relations and solve for \(S_n\).

PYQ — JEE Main 2024 (Jan 31 Shift 2)

Question: Find the sum of the series \(3 + 7 + 14 + 24 + 37 + \dots\) to 20 terms.

Solution:

Sequence: 3, 7, 14, 24, 37, ...

1st Differences: 4, 7, 10, 13, ... (This is an AP with \(a=4, d=3\)).

2nd Differences: 3, 3, 3, ... (Constant).

Since the 2nd difference is constant, the general term \(T_n\) is quadratic: \(T_n = an^2 + bn + c\).

Using \(T_1=3, T_2=7, T_3=14\): \(a+b+c=3\), \(4a+2b+c=7\), \(9a+3b+c=14\).

Solving gives \(a=3/2, b=1/2, c=1\). So \(T_n = \frac{3n^2 + n + 2}{2}\).

Sum \(S_n = \frac{3}{2}\sum n^2 + \frac{1}{2}\sum n + \sum 1 = \frac{3}{2}\frac{n(n+1)(2n+1)}{6} + \frac{1}{2}\frac{n(n+1)}{2} + n\).

Answer: Evaluate at \(n=20\)

10

Logarithmic Series

The expansion of \(\ln(1+x)\) and \(\ln(1-x)\) in powers of \(x\) yields standard infinite series.

10.1 Expansion of ln(1+x)

Most Repeated PYQ — Log Series

\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \quad (\text{for } -1 < x \le 1)

10.2 Expansion of ln(1-x)

\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \dots \quad (\text{for } -1 \le x < 1)

10.3 Useful Results

Subtracting the two:

\ln\left(\frac{1+x}{1-x}\right) = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \dots\right)

PYQ — JEE Main 2019 (Apr 10 Shift 1)

Question: The coefficient of \(x^{15}\) in the expansion of \(\ln(1+x)\) is?

Solution:

Given series: \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots + (-1)^{n-1}\frac{x^n}{n} + \dots\)

For \(x^{15}\), \(n = 15\).

Coefficient = \(\frac{(-1)^{15-1}}{15} = \frac{(-1)^{14}}{15} = \frac{1}{15}\).

Answer: \(1/15\)

11

Inequalities (AM ≥ GM)

The AM-GM inequality is heavily used in algebra to find minimum or maximum values of expressions.

11.1 Weighted AM-GM Inequality

Most Repeated PYQ — Weighted AM-GM

For \(n\) positive real numbers \(a_1, a_2, \dots, a_n\):

\frac{a_1 + a_2 + \dots + a_n}{n} \ge (a_1 \cdot a_2 \dots a_n)^{1/n}

Equality holds if all numbers are equal.

PYQ — JEE Main 2022 (July 25 Shift 1)

Question: If \(x, y, z\) are positive reals such that \(x + y + z = 15\), find the minimum value of \((1/x + 1/y + 1/z)\).

Solution:

Using AM-HM inequality: \(AM \ge HM\).

\frac{x+y+z}{3} \ge \frac{3}{1/x + 1/y + 1/z}

Given \(x+y+z=15\), we have \(\frac{15}{3} \ge \frac{3}{1/x + 1/y + 1/z}\).

5 \ge \frac{3}{1/x + 1/y + 1/z} \implies 1/x + 1/y + 1/z \ge \frac{3}{5}

The minimum value is \(3/5\) when \(x=y=z=5\).

Answer: \(3/5\)

12

Important Properties & Results

12.1 Key Formulae Summary

AP General Term: \(T_n = a + (n-1)d\)

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

GP General Term: \(T_n = ar^{n-1}\)

GP Sum: \(S_n = \frac{a(r^n - 1)}{r - 1}\)

GP Infinite Sum: \(S_\infty = \frac{a}{1-r}\)

AM-GM-HM Relation: \(G^2 = A \cdot H\) and \(A \ge G \ge H\)

AGP Infinite Sum: \(S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}\)

PYQ — JEE Main 2024 (Jan 29 Shift 1)

Question: If the sum of the first \(n\) terms of an AP is \(S_n = 3n^2 + 5n\), find the 10th term.

Solution:

We know \(T_n = S_n - S_{n-1}\).

S_{10} = 3(100) + 50 = 350

\[S_9 = 3(81) + 45 = 288\]

T_{10} = 350 - 288 = 62

Answer: 62

Previous Year Questions Analysis

JEE Main (Last 10 Years)

✓ AP/GP Formulas: 25%
✓ AM-GM-HM Inequalities: 20%
✓ Telescopic Series: 15%
✓ AGP Sums: 15%
✓ Special Series (Sum of n, n², n³): 10%
✓ Method of Differences: 10%
✓ Logarithmic Series: 5%

JEE Advanced (Last 10 Years)

✓ Multi-concept AP/GP problems: 30%
✓ Telescopic & Partial Fractions: 25%
✓ AM-GM Inequality min/max: 20%
✓ Method of Differences: 15%
✓ AGP to Infinity: 10%

Weightage Analysis

JEE Main: 4-8 marks (1-2 questions)

JEE Advanced: 4-8 marks

Difficulty Level: Easy to Moderate

Time Required: 5-6 hours practice

Practice Problem Set

Level 1: Basic (JEE Main Standard)

Find the sum of the first 20 terms of the AP: 5, 8, 11, ...
Find the sum to infinity of the GP: 1, 1/3, 1/9, ...
Insert 3 arithmetic means between 4 and 24.
Find the 10th term of the GP: 3, 6, 12, ...
Find the harmonic mean of 4 and 6.
Find the sum of \(1^2 + 2^2 + \dots + 10^2\).

Level 2: Intermediate (JEE Main/Advanced)

Find the sum to infinity of \(1 + \frac{2}{3} + \frac{3}{3^2} + \frac{4}{3^3} + \dots\)
Find the sum of \( \frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \dots \) to \(n\) terms.
Find the minimum value of \(x + \frac{1}{x}\) for \(x > 0\).
If AM and GM of two numbers are 5 and 4 respectively, find the numbers.
Find the sum of \(n\) terms of the series whose \(n\)-th term is \(n^2 + 3n\).

Level 3: Advanced (JEE Advanced/Olympiad)

Prove that \(a^2 + b^2 + c^2 \ge ab + bc + ca\) using AM-GM inequality.
Find the sum of the series \(1^3 - 2^3 + 3^3 - 4^3 + \dots\) to \(2n\) terms.
Find the sum to infinity of the series \(1 + (1+b)x + (1+b)^2 x^2 + \dots\) given \(|x(1+b)| < 1\).
If \(a, b, c, d\) are in GP, prove that \((b-c)^2 + (c-a)^2 + (d-b)^2 = (a-d)^2\).

> FAQ

Frequently Asked Questions

What is the difference between sequence and series?

A sequence is an ordered list of numbers following a specific pattern (e.g., 2, 4, 6, 8). A series is the sum of the terms of a sequence (e.g., 2 + 4 + 6 + 8).

What is the relation between AM, GM, and HM?

For any two positive numbers, AM ≥ GM ≥ HM. Also, the square of the Geometric Mean (GM) is equal to the product of the Arithmetic Mean (AM) and Harmonic Mean (HM), i.e., GM² = AM × HM.

How to find the sum of an infinite Geometric Progression (GP)?

The sum of an infinite GP with first term 'a' and common ratio 'r' (where |r| < 1) is given by S∞ = a / (1 - r). If |r| ≥ 1, the sum diverges to infinity.

What is a Telescopic Series?

A telescopic series is one where most terms cancel out when expanded. It is usually solved using partial fractions, leaving only the first and last terms, making the sum easy to calculate.

Sequences & Series notes, formulas & PYQs for JEE 2026

📑 Table of Contents

Introduction to Sequences & Series

1.1 Sequence

1.2 Series

1.3 Progression

📝 PYQ — JEE Main 2021 (Mar 17 Shift 1)

Arithmetic Progression (AP)

2.1 Standard Form & General Term

🔥 Most Repeated PYQ — General Term of AP

2.2 Sum of n Terms of AP

🔥 Most Repeated PYQ — Sum of AP

2.3 Properties of AP

2.4 Arithmetic Mean (AM)

⭐ Important — Arithmetic Mean

📝 PYQ — JEE Main 2023 (Apr 10 Shift 2)

Geometric Progression (GP)

3.1 General Term of GP

🔥 Most Repeated PYQ — General Term of GP

3.2 Sum of n Terms of GP

🔥 Most Repeated PYQ — Sum of Finite GP

3.3 Sum of Infinite GP

🔥 Most Repeated PYQ — Infinite GP

3.4 Geometric Mean (GM)

⭐ Important — Geometric Mean

3.5 Properties of GP

📝 PYQ — JEE Main 2024 (Jan 30 Shift 1)

Harmonic Progression (HP)

4.1 General Term of HP

🔥 Most Repeated PYQ — General Term of HP

4.2 Harmonic Mean (HM)

⭐ Important — Harmonic Mean

📝 PYQ — JEE Main 2020 (Sep 5 Shift 1)

Relation between AM, GM, and HM

5.1 Inequality AM ≥ GM ≥ HM

🔥 Most Repeated PYQ — Inequality

5.2 Relation AM, GM, HM

🔥 Most Repeated PYQ — AM, GM, HM Formula

5.3 General Formulas for Means

📝 PYQ — JEE Main 2022 (June 28 Shift 1)

Arithmetico-Geometric Progression (AGP)

6.1 Standard Form

🔥 Most Repeated PYQ — AGP Sequence

6.2 Sum of n Terms of AGP

🔥 Most Repeated PYQ — Sum of AGP

6.3 Sum to Infinity of AGP

🔥 Most Repeated PYQ — Infinite AGP

📝 PYQ — JEE Main 2023 (Jan 31 Shift 2)

Sum of Special Series

7.1 Sum of First n Natural Numbers

🔥 Most Repeated PYQ — Sum of Natural Numbers

7.2 Sum of Squares of First n Natural Numbers

🔥 Most Repeated PYQ — Sum of Squares

7.3 Sum of Cubes of First n Natural Numbers

🔥 Most Repeated PYQ — Sum of Cubes

📝 PYQ — JEE Main 2024 (Apr 5 Shift 2)

Telescopic Series

8.1 Partial Fractions Method

🔥 Most Repeated PYQ — Telescopic Sum

8.2 Factorial Series

📝 PYQ — JEE Main 2024 (Apr 6 Shift 1)

Method of Differences

9.1 Sum using Differences

🔥 Most Repeated PYQ — Method of Differences

📝 PYQ — JEE Main 2024 (Jan 31 Shift 2)

Logarithmic Series

10.1 Expansion of ln(1+x)

🔥 Most Repeated PYQ — Log Series

10.2 Expansion of ln(1-x)

10.3 Useful Results

📝 PYQ — JEE Main 2019 (Apr 10 Shift 1)

Inequalities (AM ≥ GM)

11.1 Weighted AM-GM Inequality

🔥 Most Repeated PYQ — Weighted AM-GM

📝 PYQ — JEE Main 2022 (July 25 Shift 1)

Important Properties & Results

12.1 Key Formulae Summary

📝 PYQ — JEE Main 2024 (Jan 29 Shift 1)

📝 Previous Year Questions Analysis

JEE Main (Last 10 Years)

Table of Contents

PYQ — JEE Main 2021 (Mar 17 Shift 1)

Most Repeated PYQ — General Term of AP

Most Repeated PYQ — Sum of AP

Important — Arithmetic Mean

PYQ — JEE Main 2023 (Apr 10 Shift 2)

Most Repeated PYQ — General Term of GP

Most Repeated PYQ — Sum of Finite GP

Most Repeated PYQ — Infinite GP

Important — Geometric Mean

PYQ — JEE Main 2024 (Jan 30 Shift 1)

Most Repeated PYQ — General Term of HP

Important — Harmonic Mean

PYQ — JEE Main 2020 (Sep 5 Shift 1)

Most Repeated PYQ — Inequality

Most Repeated PYQ — AM, GM, HM Formula

PYQ — JEE Main 2022 (June 28 Shift 1)

Most Repeated PYQ — AGP Sequence

Most Repeated PYQ — Sum of AGP

Most Repeated PYQ — Infinite AGP

PYQ — JEE Main 2023 (Jan 31 Shift 2)

Most Repeated PYQ — Sum of Natural Numbers

Most Repeated PYQ — Sum of Squares

Most Repeated PYQ — Sum of Cubes

PYQ — JEE Main 2024 (Apr 5 Shift 2)

Most Repeated PYQ — Telescopic Sum

PYQ — JEE Main 2024 (Apr 6 Shift 1)

Most Repeated PYQ — Method of Differences

PYQ — JEE Main 2024 (Jan 31 Shift 2)

Most Repeated PYQ — Log Series

PYQ — JEE Main 2019 (Apr 10 Shift 1)

Most Repeated PYQ — Weighted AM-GM

PYQ — JEE Main 2022 (July 25 Shift 1)

PYQ — JEE Main 2024 (Jan 29 Shift 1)

Previous Year Questions Analysis

Practice Problem Set

Why This Chapter is Important

Must-Remember Formulas & Values

5-Step Exam Strategy