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Sequence AND Series formulas

Master Sequence AND Series through 36 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Sequence AND Series, every formula

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nth term from Sn
$T_{n} = S_{n} - S_{n - 1}$
AP general form
$a, a + d, a + 2 d, \dots$
a = first term, d = common difference
Q1MCQAP form
In the AP $3,7,11,\dots$ the common difference is:
- A
  $4$
- B
  $3$
- C
  $7$
- D
  $11$
AP nth term
$T_{n} = a + (n - 1) d$
AP nth term from end
$T_{m}^{'} = l - (m - 1) d$
l = last term
AP sum of n terms
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$
AP three-term condition
$a, b, c in AP \Leftrightarrow 2 b = a + c$
Q1MCQAP condition
$a,b,c$ are in AP iff:
- A
  $2b=a+c$
- B
  $b^{2}=ac$
- C
  $b=\dfrac{2ac}{a+c}$
- D
  $b=a+c$
Arithmetic mean
$A = \frac{a + b}{2}$
n AMs between a,b
$d = \frac{b - a}{n + 1}$
Tn linear ⇒ AP
$T_{n} = A n + B \Rightarrow AP with d = A$
Q1MCQTn linear
If $T_n=3n+2$, the sequence is an AP with common difference:
- A
  $3$
- B
  $2$
- C
  $5$
- D
  $6$
Sn quadratic ⇒ AP
$S_{n} = A n^{2} + B n + C \Rightarrow AP with d = 2 A$
Q1MCQSn quadratic
If $S_n=2n^{2}+3n$, the common difference of the AP is:
- A
  $4$
- B
  $2$
- C
  $3$
- D
  $5$
GP general form
$a, a r, a r^{2}, \dots$
a = first term, r = common ratio
Q1MCQGP form
In the GP $3,6,12,\dots$ the common ratio is:
- A
  $2$
- B
  $3$
- C
  $\dfrac12$
- D
  $6$
GP nth term
$T_{n} = a r^{n - 1}$
GP sum of n terms
$S_{n} = a \frac{r ^{n} - 1}{r - 1}, r \neq = 1$
GP infinite sum
$S_{\infty} = \frac{a}{1 - r}, ∣ r ∣ < 1$
GP three-term condition
$a, b, c in GP \Leftrightarrow b^{2} = a c$
Q1MCQGP condition
$a,b,c$ are in GP iff:
- A
  $b^{2}=ac$
- B
  $2b=a+c$
- C
  $b=\dfrac{2ac}{a+c}$
- D
  $b=ac$
Geometric mean
$G = ab$
n GMs between a,b
$r = (\frac{b}{a})^{1/ (n + 1)}$
Product of n GMs
$G_{1} G_{2} \dots G_{n} = (ab)^{n}$
Q1MCQProduct of GMs
The product of $n$ GMs between $a$ and $b$ equals:
- A
  $(\sqrt{ab})^{n}$
- B
  $\sqrt{ab}$
- C
  $ab$
- D
  $(ab)^{n}$
HP form
$\frac{1}{a}, \frac{1}{a + d}, \frac{1}{a + 2 d}, \dots$
reciprocals in AP
Q1MCQHP form
A sequence is in HP if the reciprocals of its terms are in:
- A
  AP
- B
  GP
- C
  HP
- D
  AGP
HP three-term condition
$a, b, c in HP \Leftrightarrow b = \frac{2 a c}{a + c}$
Harmonic mean
$H = \frac{2 ab}{a + b}$
AM–GM–HM inequality
$A \geq G \geq H$
positive numbers
Q1MCQAM-GM-HM
For positive numbers, the correct ordering is:
- A
  $A\ge G\ge H$
- B
  $H\ge G\ge A$
- C
  $G\ge A\ge H$
- D
  $A=G=H$ always
A,G,H relation
$G^{2} = A H$
Q1MCQA,G,H relation
For two numbers, $G^{2}$ equals:
- A
  $AH$
- B
  $A+H$
- C
  $\dfrac{A}{H}$
- D
  $A-H$
AGP nth term
$T_{n} = [a + (n - 1) d] r^{n - 1}$
Q1MCQAGP nth term
The $n$th term of an AGP is:
- A
  $[a+(n-1)d]\,r^{\,n-1}$
- B
  $a+(n-1)d$
- C
  $ar^{n-1}$
- D
  $a+(n-1)dr$
AGP infinite sum
$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}, ∣ r ∣ < 1$
Q1MCQAGP infinite
$S_\infty$ of an AGP (with $|r|<1$) is:
- A
  $\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^{2}}$
- B
  $\dfrac{a}{1-r}$
- C
  $\dfrac{dr}{(1-r)^{2}}$
- D
  $\dfrac{a}{(1-r)^{2}}$
Sum of first n naturals
$k = 1 \sum n k = \frac{n ( n + 1 )}{2}$
Sum of squares
$k = 1 \sum n k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$
Sum of cubes
$k = 1 \sum n k^{3} = (\frac{n ( n + 1 )}{2})^{2}$
Sum of odd numbers
$1 + 3 + 5 + \dots + (2 n - 1) = n^{2}$
Sum of even numbers
$2 + 4 + 6 + \dots + 2 n = n (n + 1)$
Cauchy–Schwarz
$(\sum a_{i} b_{i})^{2} \leq (\sum a_{i}^{2}) (\sum b_{i}^{2})$
Q1MCQCauchy–Schwarz
Cauchy–Schwarz states $\left(\sum a_ib_i\right)^{2}$ is:
- A
  $\le\left(\sum a_i^{2}\right)\left(\sum b_i^{2}\right)$
- B
  $\ge\left(\sum a_i^{2}\right)\left(\sum b_i^{2}\right)$
- C
  $=\sum a_i^{2}+\sum b_i^{2}$
- D
  $=\sum a_ib_i$
(1-x)^{-1} series
$(1 - x)^{- 1} = 1 + x + x^{2} + \dots, ∣ x ∣ < 1$
Q1MCQ(1-x)^{-1}
For $|x|<1$, $(1-x)^{-1}$ equals:
- A
  $1+x+x^{2}+\cdots$
- B
  $1-x+x^{2}-\cdots$
- C
  $1+2x+3x^{2}+\cdots$
- D
  $1-x^{2}+\cdots$
(1-x)^{-2} series
$(1 - x)^{- 2} = 1 + 2 x + 3 x^{2} + \dots, ∣ x ∣ < 1$
Q1MCQ(1-x)^{-2}
For $|x|<1$, the coefficient of $x^{2}$ in $(1-x)^{-2}$ is:
- A
  $3$
- B
  $2$
- C
  $1$
- D
  $4$
Equidistant terms in AP
$a_{2} + a_{n - 1} = a_{1} + a_{n}$
Q1MCQEquidistant terms
In an AP, $a_2+a_{n-1}$ equals:
- A
  $a_1+a_n$
- B
  $2a_1$
- C
  $a_n$
- D
  $a_1a_n$
p-th=q, q-th=p ⇒
$T_{p} = q, T_{q} = p \Rightarrow T_{p + q} = 0$
Sp=q, Sq=p ⇒
$S_{p} = q, S_{q} = p \Rightarrow S_{p + q} = - (p + q)$
Q1MCQSp=q etc.
If $S_p=q$ and $S_q=p$ in an AP, then $S_{p+q}$ equals:
- A
  $-(p+q)$
- B
  $p+q$
- C
  $0$
- D
  $pq$

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Sequence AND Series formulas

Sequence AND Series, every formula

nth term from Sn

AP general form

AP nth term

AP nth term from end

AP sum of n terms

AP three-term condition

Arithmetic mean

n AMs between a,b

Tn linear ⇒ AP

Sn quadratic ⇒ AP

GP general form

GP nth term

GP sum of n terms

GP infinite sum

GP three-term condition

Geometric mean

n GMs between a,b

Product of n GMs

HP form

HP three-term condition

Harmonic mean

AM–GM–HM inequality

A,G,H relation

AGP nth term

AGP infinite sum

Sum of first n naturals

Sum of squares

Sum of cubes

Sum of odd numbers

Sum of even numbers

Cauchy–Schwarz

(1-x)^{-1} series

(1-x)^{-2} series

Equidistant terms in AP

p-th=q, q-th=p ⇒

Sp=q, Sq=p ⇒

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