Sequences and Series Formula Sheet

Every must-know formula for Sequences and Series in one scannable sheet, grouped by progression type. Use it for last-minute revision before JEE Main and Advanced.

Arithmetic Progression (AP)

General form: $a,\ a+d,\ a+2d,\ a+3d,\ \ldots$ where $a$ is the first term and $d$ is the common difference.

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

where $l = T_n$ is the last term.

Quantity	Formula	Notes
$n$th term	$T_n = a + (n-1)d$	Not $a + nd$ — first term is $a$
Sum of $n$ terms	$S_n = \frac{n}{2}[2a + (n-1)d]$	Note the $2a$, not $a$
Sum (last term known)	$S_n = \frac{n}{2}(a + l)$	Count $\times$ average of ends
AP test	$T_{n+1} - T_n = $ constant	Or $2b = a + c$ for $a,b,c$
$n$th term from sum	$T_n = S_n - S_{n-1}$	Valid for $n \geq 2$

Key Properties

• Three terms in AP: $$\boxed{2b = a + c}$$
• Sum of equidistant terms: $T_k + T_{n-k+1} = a + l$ (constant)
• Arithmetic Mean of $a$ and $b$: $A = \dfrac{a+b}{2}$
• Inserting $n$ AMs between $a$ and $b$: common difference $d = \dfrac{b-a}{n+1}$
• $S_n,\ S_{2n} - S_n,\ S_{3n} - S_{2n},\ \ldots$ are themselves in AP

Smart Selection of Terms

Terms	Choose as	Sum
3 terms	$a-d,\ a,\ a+d$	$3a$
4 terms	$a-3d,\ a-d,\ a+d,\ a+3d$	$4a$
5 terms	$a-2d,\ a-d,\ a,\ a+d,\ a+2d$	$5a$

Geometric Progression (GP)

General form: $a,\ ar,\ ar^2,\ ar^3,\ \ldots$ with first term $a \neq 0$ and common ratio $r \neq 0$.

$$\boxed{T_n = ar^{n-1}}$$$$\boxed{S_n = a\frac{r^n - 1}{r - 1} = a\frac{1 - r^n}{1 - r}\quad (r \neq 1)}$$

Quantity	Formula	Notes
$n$th term	$T_n = ar^{n-1}$	Not $ar^n$
Sum ($r > 1$)	$S_n = a\frac{r^n - 1}{r - 1}$	Positive denominator
Sum ($r < 1$)	$S_n = a\frac{1 - r^n}{1 - r}$	Positive denominator
Sum ($r = 1$)	$S_n = na$	Formula above is undefined
Common ratio	$r = \frac{T_{n+1}}{T_n}$	Constant for a GP

Key Properties

• Three terms in GP: $$\boxed{b^2 = ac}$$
• Product of equidistant terms: $T_k \cdot T_{n-k+1} = a \cdot l$ (constant)
• Geometric Mean of $a$ and $b$: $G = \sqrt{ab}$
• Inserting $n$ GMs between $a$ and $b$: ratio $r = \left(\dfrac{b}{a}\right)^{\frac{1}{n+1}}$
• Reciprocals of GP terms form a GP with ratio $\frac{1}{r}$
• $\frac{T_8}{T_5} = r^3$ — divide terms to get $r$ raised to the index gap

Smart Selection of Terms

Terms	Choose as	Product
3 terms	$\frac{a}{r},\ a,\ ar$	$a^3$
4 terms	$\frac{a}{r^3},\ \frac{a}{r},\ ar,\ ar^3$	$a^4$
5 terms	$\frac{a}{r^2},\ \frac{a}{r},\ a,\ ar,\ ar^2$	$a^5$

Infinite GP

Converges if and only if $|r| < 1$.

$$\boxed{S_\infty = \frac{a}{1-r}\quad (|r| < 1)}$$

Quantity	Formula	Notes
Infinite sum	$S_\infty = \frac{a}{1-r}$	Only when $\lvert r\rvert < 1$
Convergence	$\lvert r\rvert < 1$	Use absolute value; $1-r$ in denominator, not $1-\lvert r\rvert$
Pure recurring	$0.\overline{a_1\ldots a_k} = \frac{a_1\ldots a_k}{\underbrace{99\ldots9}_{k}}$	$k$ digits $\to k$ nines
Mixed recurring	$0.a_1\ldots a_m\overline{b_1\ldots b_k} = \frac{a_1\ldots a_m b_1\ldots b_k - a_1\ldots a_m}{\underbrace{9\ldots9}_{k}\underbrace{0\ldots0}_{m}}$	$k$ nines then $m$ zeros

Standard infinite-series results (for $|x| < 1$):

$$1 - x + x^2 - x^3 + \cdots = \frac{1}{1+x}\qquad (\text{ratio } -x)$$$$0.\overline{9} = 1 \quad\text{(exactly)}$$

Harmonic Progression (HP)

A sequence whose reciprocals form an AP. If the reciprocal AP has first term $a$ and common difference $d$:

$$\boxed{H_n = \frac{1}{a + (n-1)d}}$$

Quantity	Formula	Notes
$n$th term	$H_n = \frac{1}{a+(n-1)d}$	Work with reciprocal AP
Reciprocal relation	$\frac{1}{H_n} = \frac{1}{H_1} + (n-1)d$	$d = \frac{1}{H_2} - \frac{1}{H_1}$
Three terms in HP	$\frac{2}{b} = \frac{1}{a} + \frac{1}{c}$	Equivalent to $b = \frac{2ac}{a+c}$
Harmonic Mean (2 nos.)	$H = \frac{2ab}{a+b}$	Not $\frac{a+b}{2ab}$
Harmonic Mean ($n$ nos.)	$H = \frac{n}{\frac{1}{a_1} + \cdots + \frac{1}{a_n}}$	Reciprocal of mean of reciprocals

• Three terms in HP: $$\boxed{b = \frac{2ac}{a+c}}$$
• Selection of 3 terms: $\frac{1}{a-d},\ \frac{1}{a},\ \frac{1}{a+d}$ (reciprocals of an AP)

AM, GM, HM and the Mean Inequality

For positive numbers $a$ and $b$:

Mean	Two numbers	$n$ numbers
Arithmetic (AM)	$\frac{a+b}{2}$	$\frac{a_1 + \cdots + a_n}{n}$
Geometric (GM)	$\sqrt{ab}$	$\sqrt[n]{a_1 a_2 \cdots a_n}$
Harmonic (HM)	$\frac{2ab}{a+b}$	$\frac{n}{\frac{1}{a_1} + \cdots + \frac{1}{a_n}}$

$$\boxed{\text{AM} \geq \text{GM} \geq \text{HM}}$$$$\boxed{\frac{a+b}{2} \geq \sqrt{ab} \geq \frac{2ab}{a+b}}$$

Equality holds if and only if all numbers are equal ($a = b$).

Key Relations

$$\boxed{\text{GM}^2 = \text{AM} \times \text{HM}}$$

• Standard optimization result: $$\boxed{x + \frac{k}{x} \geq 2\sqrt{k}\quad (x > 0)}$$(so $x + \frac{1}{x} \geq 2$)
• Weighted AM-GM: $\dfrac{ma + nb}{m+n} \geq \sqrt[m+n]{a^m b^n}$, equality when $a = b$

Advanced Results (JEE Advanced)

• Power Mean Inequality (for $r > s$): $$\left(\frac{a_1^r + \cdots + a_n^r}{n}\right)^{1/r} \geq \left(\frac{a_1^s + \cdots + a_n^s}{n}\right)^{1/s}$$ ($r=1, s=0$ gives AM $\geq$ GM; $r=0, s=-1$ gives GM $\geq$ HM)
• Cauchy-Schwarz Inequality: $$(a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2) \geq (a_1 b_1 + \cdots + a_n b_n)^2$$

Relationship Between AP, GP, HP

For a number $b$ between $a$ and $c$:

graph LR
    A["a, b, c in AP
b = (a+c)/2"] --> M["AM ≥ GM ≥ HM
(positive a, c)"]
    G["a, b, c in GP
b = √(ac)"] --> M
    H["a, b, c in HP
b = 2ac/(a+c)"] --> M

Arithmetico-Geometric Progression (AGP)

Each term is the product of an AP term and a GP term: $a,\ (a+d)r,\ (a+2d)r^2,\ \ldots$

$$\boxed{T_n = \big[a + (n-1)d\big]\, r^{n-1}}$$$$\boxed{S_n = \frac{a}{1-r} + \frac{dr\,(1 - r^{n-1})}{(1-r)^2} - \frac{[a+(n-1)d]\,r^n}{1-r}\quad (r \neq 1)}$$

When $r = 1$, the AGP reduces to an AP: $S_n = na + \frac{n(n-1)d}{2}$.

Infinite AGP ($|r| < 1$)

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

Must-Know Standard Sums ($|x| < 1$)

Series	Sum
$1 + 2x + 3x^2 + 4x^3 + \cdots$	$\frac{1}{(1-x)^2}$
$x + 2x^2 + 3x^3 + 4x^4 + \cdots$	$\frac{x}{(1-x)^2}$
$1 + 4x + 9x^2 + 16x^3 + \cdots$	$\frac{1+x}{(1-x)^3}$
$a + (a+d)r + (a+2d)r^2 + \cdots$	$\frac{a + r(d-a)}{(1-r)^2}$

Special Series and Standard Summations

$$\boxed{\sum_{k=1}^{n} k = \frac{n(n+1)}{2}}$$$$\boxed{\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}}$$$$\boxed{\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2}$$

Series	Sum	Notes
$\sum k$	$\frac{n(n+1)}{2}$	Natural numbers
$\sum k^2$	$\frac{n(n+1)(2n+1)}{6}$	Three factors, $\div 6$
$\sum k^3$	$\left[\frac{n(n+1)}{2}\right]^2$	Square of $\sum k$; $\div 4$
$\sum 2k$	$n(n+1)$	First $n$ even numbers
$\sum (2k-1)$	$n^2$	First $n$ odd numbers
$\sum k(k+1)$	$\frac{n(n+1)(n+2)}{3}$	Product of 2 consecutive
$\sum k(k+1)(k+2)$	$\frac{n(n+1)(n+2)(n+3)}{4}$	Product of 3 consecutive
$\sum k^4$	$\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$	Rare in JEE

Useful Combinations and Patterns

• Linear combination: $\sum (ak + b) = a\dfrac{n(n+1)}{2} + bn$
• General product rule: $$\boxed{\sum_{k=1}^{n} k(k+1)\cdots(k+r-1) = \frac{n(n+1)\cdots(n+r)}{r+1}}$$
• Range adjustment: $\displaystyle\sum_{k=a}^{b} f(k) = \sum_{k=1}^{b} f(k) - \sum_{k=1}^{a-1} f(k)$
• Alternating sum: $1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n = \begin{cases} \frac{n+1}{2} & n \text{ odd} \\ -\frac{n}{2} & n \text{ even} \end{cases}$

Method of Differences (Telescoping)

If $T_n = f(n+1) - f(n)$, then almost everything cancels:

$$\boxed{S_n = \sum_{k=1}^{n} \big[f(k+1) - f(k)\big] = f(n+1) - f(1)}$$

Standard Telescoping Forms

Series	Sum	Key step
$\sum \frac{1}{k(k+1)}$	$\frac{n}{n+1}$	$\frac{1}{k} - \frac{1}{k+1}$
$\sum \frac{1}{k(k+2)}$	$\frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$	$\frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)$
$\sum \frac{1}{k(k+1)(k+2)}$	$\frac{1}{4} - \frac{1}{2(n+1)(n+2)}$	Difference of products
$\sum \frac{1}{(2k-1)(2k+1)}$	$\frac{n}{2n+1}$	$\frac{1}{2}\left(\frac{1}{2k-1} - \frac{1}{2k+1}\right)$
$\sum \frac{1}{\sqrt{k} + \sqrt{k+1}}$	$\sqrt{n+1} - 1$	Rationalize: $\sqrt{k+1} - \sqrt{k}$
$\sum k\cdot k!$	$(n+1)! - 1$	$k\cdot k! = (k+1)! - k!$
$\sum \frac{k}{(k+1)!}$	$1 - \frac{1}{(n+1)!}$	$\frac{1}{k!} - \frac{1}{(k+1)!}$

Cross-Links

• Arithmetic Progression
• Geometric Progression
• Infinite GP
• Harmonic Progression
• AM-GM-HM Inequalities
• AGP
• Special Series
• Method of Differences