Sequence AND Series formulas

Every formula JEE Advanced expects you to know for Sequence AND Series, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 36
Subject: Mathematics
Exam: JEE Advanced
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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
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
$a, b, c in AP \Leftrightarrow 2 b = a + c$
Arithmetic mean
$A = \frac{a + b}{2}$
n AMs between a,b
$d = \frac{b - a}{n + 1}$
Tn linear ⇒ AP
$T_{n} = A n + B \Rightarrow AP with d = A$
Sn quadratic ⇒ AP
$S_{n} = A n^{2} + B n + C \Rightarrow AP with d = 2 A$
GP general form
$a, a r, a r^{2}, \dots$
a = first term, r = common ratio
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
$a, b, c in GP \Leftrightarrow b^{2} = a c$
Geometric mean
$G = ab$
n GMs between a,b
$r = (\frac{b}{a})^{1/ (n + 1)}$
Product of n GMs
$G_{1} G_{2} \dots G_{n} = (ab)^{n}$
HP form
$\frac{1}{a}, \frac{1}{a + d}, \frac{1}{a + 2 d}, \dots$
reciprocals in AP
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
A,G,H relation
$G^{2} = A H$
AGP nth term
$T_{n} = [a + (n - 1) d] r^{n - 1}$
AGP infinite sum
$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}, ∣ r ∣ < 1$
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
$(\sum a_{i} b_{i})^{2} \leq (\sum a_{i}^{2}) (\sum b_{i}^{2})$
(1-x)^{-1} series
$(1 - x)^{- 1} = 1 + x + x^{2} + \dots, ∣ x ∣ < 1$
(1-x)^{-2} series
$(1 - x)^{- 2} = 1 + 2 x + 3 x^{2} + \dots, ∣ x ∣ < 1$
Equidistant terms in AP
$a_{2} + a_{n - 1} = a_{1} + a_{n}$
p-th=q, q-th=p ⇒
$T_{p} = q, T_{q} = p \Rightarrow T_{p + q} = 0$
Sp=q, Sq=p ⇒
$S_{p} = q, S_{q} = p \Rightarrow S_{p + q} = - (p + q)$

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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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