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Sequences & Series: Your Final Revision Sheet & Must-Know Formulas

Everything you need to master Sequences & Series for JEE Main & Advanced - from basic definitions to advanced problem-solving strategies.

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Arithmetic Progression Geometric Progression Harmonic Progression Special Series Arithmetic Mean Summation Formulas JEE Strategies

Why Sequences & Series Matter in JEE

Sequences & Series contribute 4-8 marks in every JEE paper and form the foundation for many advanced mathematical concepts. Mastering this topic gives you:

Quick, guaranteed marks in objective-type questions
Strong foundation for calculus and algebra
Problem-solving patterns applicable across mathematics
Time efficiency - most problems can be solved in 2-3 minutes

1. Arithmetic Progression (AP)

General Form & nth Term

General AP: $a, a+d, a+2d, a+3d, \dots$

nth term: $T_n = a + (n-1)d$

Where: $a$ = first term, $d$ = common difference

Sum of First n Terms

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

Alternative form: $S_n = \frac{n}{2}(a + l)$ where $l$ = last term

Relation: $T_n = S_n - S_{n-1}$

Properties of AP

Three terms in AP: $a-d, a, a+d$

Four terms in AP: $a-3d, a-d, a+d, a+3d$

Five terms in AP: $a-2d, a-d, a, a+d, a+2d$

Arithmetic Mean: $AM = \frac{a+b}{2}$

💡 AP Problem-Solving Tips

If sum of n terms is given as $S_n = An^2 + Bn$, then $d = 2A$ and $a = A + B$
For symmetric terms in AP, use the property: $T_k + T_{n-k+1} = \text{constant}$
If $p$th term = $q$ and $q$th term = $p$, then $(p+q)$th term = $0$

2. Geometric Progression (GP)

General Form & nth Term

General GP: $a, ar, ar^2, ar^3, \dots$

nth term: $T_n = ar^{n-1}$

Where: $a$ = first term, $r$ = common ratio

Sum of First n Terms

When |r| < 1: $S_n = \frac{a(1-r^n)}{1-r}$

When |r| > 1: $S_n = \frac{a(r^n-1)}{r-1}$

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

Properties of GP

Three terms in GP: $\frac{a}{r}, a, ar$

Four terms in GP: $\frac{a}{r^3}, \frac{a}{r}, ar, ar^3$

Geometric Mean: $GM = \sqrt{ab}$

Product of n terms: $P_n = a^n r^{\frac{n(n-1)}{2}}$

⚠️ Common GP Mistakes

Using infinite GP formula when |r| ≥ 1
Forgetting that common ratio can be negative
Mishandling product of terms in GP
Confusing AM and GM inequality conditions

3. Harmonic Progression (HP)

HP Definition & nth Term

General HP: $\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \dots$

Reciprocal forms AP: $a, a+d, a+2d, \dots$

nth term: $T_n = \frac{1}{a+(n-1)d}$

Harmonic Mean: $HM = \frac{2ab}{a+b}$

Relation between AM, GM, HM

For two numbers: $AM \geq GM \geq HM$

Equality: When $a = b$

Relation: $(GM)^2 = AM \times HM$

For n numbers: $AM \geq GM \geq HM$

💡 HP Problem Approach

Always convert HP to AP by taking reciprocals
Solve the AP problem, then convert back to HP
Remember: No direct sum formula for HP - work with corresponding AP

4. Arithmetic & Geometric Means

n Arithmetic Means

Between a and b: $A_k = a + k\left(\frac{b-a}{n+1}\right)$

Common difference: $d = \frac{b-a}{n+1}$

Sum of n AMs: $S = n \times \frac{a+b}{2}$

n Geometric Means

Between a and b: $G_k = a\left(\frac{b}{a}\right)^{\frac{k}{n+1}}$

Common ratio: $r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

Product of n GMs: $P = (\sqrt{ab})^n$

5. Special Series & Summation Formulas

Series Type	Sum Formula	Check
Sum of first n natural numbers	$\sum_{k=1}^n k = \frac{n(n+1)}{2}$
Sum of squares of first n natural numbers	$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$
Sum of cubes of first n natural numbers	$\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2$
Sum of first n odd numbers	$\sum_{k=1}^n (2k-1) = n^2$
Sum of first n even numbers	$\sum_{k=1}^n 2k = n(n+1)$
Arithmetic-Geometric Series	$\sum_{k=1}^n (a+kd)r^{k-1} = \frac{a}{1-r} + \frac{rd(1-r^{n-1})}{(1-r)^2}$

6. Advanced Summation Techniques

Method of Differences

Telescoping Series: $\sum_{k=1}^n [f(k) - f(k+1)] = f(1) - f(n+1)$

When $T_k = f(k) - f(k+1)$, sum telescopes

Example: $\sum \frac{1}{k(k+1)} = \sum \left(\frac{1}{k} - \frac{1}{k+1}\right)$

Vn Method

For series where $T_k$ is product of factors

Find V_k such that $T_k = V_k - V_{k-1}$

Sum = $V_n - V_0$

7. JEE Problem-Solving Strategies

Quick Identification Techniques

If terms are equally spaced → AP
If ratio of consecutive terms is constant → GP
If reciprocals form AP → HP
If terms involve both addition and multiplication → AGP

Time-Saving Shortcuts

For sum of series, check if it's standard type first
Use symmetric terms property in AP: $T_k + T_{n-k+1} = \text{constant}$
For product in GP: $T_k \times T_{n-k+1} = \text{constant}$
Remember: Sum of infinite GP exists only when |r| < 1

Common Pitfalls to Avoid

Applying infinite GP formula when |r| ≥ 1
Forgetting to check domain in logarithmic sequences
Mishandling negative common ratios in GP
Confusing AM-GM-HM inequality conditions

📝 Quick Revision Test

Solve these typical JEE problems to test your understanding:

1. If the sum of n terms of an AP is $3n^2 + 5n$, find its 15th term.

Hint: Use $T_n = S_n - S_{n-1}$

2. Find the sum of the series: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ to infinity

Hint: Infinite GP with |r| < 1

3. Insert 3 arithmetic means between 3 and 19.

Hint: Total terms = 5, find common difference

Final Formula Cheat Sheet

🎯 AP Formulas

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

🚀 GP Formulas

$T_n = ar^{n-1}$
$S_n = \frac{a(1-r^n)}{1-r}$ (|r| < 1)
$S_\infty = \frac{a}{1-r}$ (|r| < 1)
$GM = \sqrt{ab}$

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