Mathematics · JEE

Sequence and Series Formula Sheet for JEE

45+ JEE formulas in this unit

Quick answer

The Sequence and Series JEE formula sheet lists 45+ important formulas for JEE Main and Advanced, including essential identities from Arithmetic and Geometric progressions, Insertion of arithmetic, geometric means between two given numbers, Relation between A.M and G.M. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Sequence and Series. This unit-wise formula list covers 45+ exam-relevant results across Arithmetic and Geometric progressions, Insertion of arithmetic, geometric means between two given numbers, Relation between A.M and G.M, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

45 formulas across 3 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 21Important: 21Supplementary: 3

Arithmetic and Geometric progressions

Arithmetic progression nth termEssential

a_n=a+(n-1)d

Variables: $a$ : first term, $d$ : common difference, $n$ : term number, $a_n$ : nth term.
Conditions: For an arithmetic progression.
Where used in JEE: Finding a general term, identifying AP, solving term-based equations.

Arithmetic progression from mth termImportant

a_n=a_m+(n-m)d

Variables: $a_n,a_m$ : nth and mth terms, $d$ : common difference.
Conditions: For an arithmetic progression.
Where used in JEE: Relating distant terms directly without first finding the first term.

Common difference of APEssential

d=a_{n+1}-a_n

Variables: $d$ : common difference, $a_n$ : nth term.
Conditions: Constant for all consecutive terms in an AP.
Where used in JEE: Testing whether a sequence is an AP, finding missing terms.

Sum of first n terms of APEssential

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

Variables: $S_n$ : sum of first $n$ terms, $a$ : first term, $d$ : common difference.
Conditions: For an arithmetic progression.
Where used in JEE: Sum problems, parameter-based AP questions, word problems.

Alternative sum formula for APEssential

S_n=\frac{n}{2}(a+l)

Variables: $S_n$ : sum of first $n$ terms, $a$ : first term, $l$ : nth or last term.
Conditions: For an arithmetic progression with last term $l=a_n$ .
Where used in JEE: When first and last terms are known.

Last term of APImportant

l=a+(n-1)d

Variables: $l$ : last term, $a$ : first term, $d$ : common difference.
Conditions: For $n$ terms of an AP.
Where used in JEE: Finite AP questions, sum using first and last terms.

Sum of AP from pth to qth termImportant

a_p+a_{p+1}+\cdots+a_q=\frac{q-p+1}{2}(a_p+a_q)

Variables: $a_p,a_q$ : pth and qth terms.
Conditions: For an arithmetic progression, $q\ge p$ .
Where used in JEE: Partial sum between two indices.

Middle term in APImportant

a_n=\frac{a_{n-r}+a_{n+r}}{2}

Variables: $a_n$ : middle term, $r$ : equal offset.
Conditions: Indices valid in the same AP.
Where used in JEE: Symmetric-term problems in AP.

Equidistant AP terms sumImportant

a_{m-r}+a_{m+r}=2a_m

Variables: $a_m$ : middle term, $r$ : equal offset.
Conditions: Indices valid in the same AP.
Where used in JEE: Simplifying sums of terms equally spaced about a middle term.

Three numbers in APEssential

2b=a+c

Variables: $a,b,c$ : three consecutive terms in AP.
Conditions: When $a,b,c$ are in arithmetic progression.
Where used in JEE: Forming numbers in AP, parameterization, equation setup.

nth term of GPEssential

a_n=ar^{n-1}

Variables: $a$ : first term, $r$ : common ratio, $n$ : term number, $a_n$ : nth term.
Conditions: For a geometric progression.
Where used in JEE: General term of GP, identifying GP, solving term equations.

Geometric progression from mth termImportant

a_n=a_m\,r^{\,n-m}

Variables: $a_n,a_m$ : nth and mth terms, $r$ : common ratio.
Conditions: For a geometric progression.
Where used in JEE: Relating distant GP terms directly.

Common ratio of GPEssential

r=\frac{a_{n+1}}{a_n}

Variables: $r$ : common ratio, $a_n$ : nth term.
Conditions: $a_n\ne 0$ ; constant for all consecutive terms in a GP.
Where used in JEE: Testing whether a sequence is a GP, finding missing terms.

Sum of first n terms of GP for r not equal to 1Essential

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

Variables: $S_n$ : sum of first $n$ terms, $a$ : first term, $r$ : common ratio.
Conditions: For a geometric progression, $r\ne 1$ .
Where used in JEE: Finite GP sums, applications in growth/decay and algebraic manipulations.

Sum of first n terms of GP for r equals 1Important

S_n=na

Variables: $S_n$ : sum of first $n$ terms, $a$ : common term.
Conditions: For a geometric progression with $r=1$ .
Where used in JEE: Special-case GP sum.

Infinite GP sumEssential

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

Variables: $S_\infty$ : sum to infinity, $a$ : first term, $r$ : common ratio.
Conditions: Valid only when $|r|<1$ .
Where used in JEE: Infinite series, recurring decimals, convergence-based JEE problems.

Convergence condition for infinite GPImportant

\lim_{n\to\infty}r^n=0\iff |r|<1

Variables: $r$ : common ratio.
Conditions: Real $r$ .
Where used in JEE: Checking existence of GP sum to infinity.

Middle term in GPImportant

a_n^2=a_{n-r}a_{n+r}

Variables: $a_n$ : middle term, $r$ : equal offset.
Conditions: Indices valid in the same GP.
Where used in JEE: Symmetric-term problems in GP.

Equidistant GP terms productImportant

a_{m-r}a_{m+r}=a_m^2

Variables: $a_m$ : middle term, $r$ : equal offset.
Conditions: Indices valid in the same GP.
Where used in JEE: Simplifying products of terms equally spaced about a middle term.

Three numbers in GPEssential

b^2=ac

Variables: $a,b,c$ : three consecutive terms in GP.
Conditions: When $a,b,c$ are in geometric progression.
Where used in JEE: Forming numbers in GP, parameterization, equation setup.

Product of first n terms of GPSupplementary

a\cdot ar\cdot ar^2\cdots ar^{n-1}=a^n r^{\frac{n(n-1)}{2}}

Variables: $a$ : first term, $r$ : common ratio, $n$ : number of terms.
Conditions: For a geometric progression.
Where used in JEE: Product-type GP problems.

Sum of first n natural numbersImportant

1+2+3+\cdots+n=\frac{n(n+1)}{2}

Variables: $n$ : positive integer.
Conditions: $n\in\mathbb{N}$ .
Where used in JEE: Standard AP sum result and simplification in sequence problems.

Sum of first n odd natural numbersImportant

1+3+5+\cdots+(2n-1)=n^2

Variables: $n$ : positive integer.
Conditions: $n\in\mathbb{N}$ .
Where used in JEE: Special AP sum result.

Sum of first n even natural numbersSupplementary

2+4+6+\cdots+2n=n(n+1)

Variables: $n$ : positive integer.
Conditions: $n\in\mathbb{N}$ .
Where used in JEE: Special AP sum result.

General AP determined by two termsImportant

d=\frac{a_q-a_p}{q-p},\quad a=a_p-(p-1)d

Variables: $a$ : first term, $d$ : common difference, $a_p,a_q$ : pth and qth terms.
Conditions: For an AP with $p\ne q$ .
Where used in JEE: Reconstructing an AP from two given terms.

General GP determined by two termsImportant

r=\left(\frac{a_q}{a_p}\right)^{\!\frac{1}{q-p}},\quad a=\frac{a_p}{r^{p-1}}

Variables: $a$ : first term, $r$ : common ratio, $a_p,a_q$ : pth and qth terms.
Conditions: For a GP with $p\ne q$ , with ratio defined appropriately in the real domain.
Where used in JEE: Reconstructing a GP from two given terms.

Insertion of arithmetic, geometric means between two given numbers

n arithmetic means between two numbersEssential

d=\frac{b-a}{n+1}

Variables: $a,b$ : given numbers, $n$ : number of arithmetic means, $d$ : common difference of resulting AP.
Conditions: When $a, A_1, A_2,\dots,A_n, b$ are in AP.
Where used in JEE: Insertion of arithmetic means.

kth arithmetic mean between two numbersEssential

A_k=a+kd=a+\frac{k(b-a)}{n+1}

Variables: $a,b$ : given numbers, $n$ : number of arithmetic means inserted, $k=1,2,\dots,n$ , $A_k$ : kth arithmetic mean.
Conditions: When $a, A_1,\dots,A_n,b$ form an AP.
Where used in JEE: Finding specific inserted arithmetic means.

Single arithmetic mean between two numbersEssential

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

Variables: $a,b$ : given numbers, $A$ : arithmetic mean.
Conditions: For one arithmetic mean between $a$ and $b$ .
Where used in JEE: Basic mean insertion and midpoint in AP.

n geometric means between two numbersEssential

r=\left(\frac{b}{a}\right)^{\!\frac{1}{n+1}}

Variables: $a,b$ : given numbers, $n$ : number of geometric means, $r$ : common ratio of resulting GP.
Conditions: When $a, G_1, G_2,\dots,G_n,b$ are in GP; ratio defined in the real domain.
Where used in JEE: Insertion of geometric means.

kth geometric mean between two numbersEssential

G_k=ar^k=a\left(\frac{b}{a}\right)^{\!\frac{k}{n+1}}

Variables: $a,b$ : given numbers, $n$ : number of geometric means inserted, $k=1,2,\dots,n$ , $G_k$ : kth geometric mean.
Conditions: When $a, G_1,\dots,G_n,b$ form a GP; expression real where defined.
Where used in JEE: Finding specific inserted geometric means.

Single geometric mean between two numbersEssential

G=\sqrt{ab}

Variables: $a,b$ : given numbers, $G$ : geometric mean.
Conditions: For real geometric mean, usually $a,b\ge 0$ .
Where used in JEE: Basic mean insertion, three numbers in GP.

Arithmetic means form an APImportant

a, A_1, A_2,\dots,A_n,b\text{ are in AP}\iff A_k=a+\frac{k(b-a)}{n+1}

Variables: $a,b$ : end terms, $A_k$ : inserted arithmetic means.
Conditions: $k=1,2,\dots,n$ .
Where used in JEE: Recognizing and generating inserted arithmetic means.

Geometric means form a GPImportant

a, G_1, G_2,\dots,G_n,b\text{ are in GP}\iff G_k=a\left(\frac{b}{a}\right)^{\!\frac{k}{n+1}}

Variables: $a,b$ : end terms, $G_k$ : inserted geometric means.
Conditions: $k=1,2,\dots,n$ ; ratio defined in the real domain.
Where used in JEE: Recognizing and generating inserted geometric means.

Relation between A.M and G.M

AM-GM inequality for two non-negative numbersEssential

\frac{a+b}{2}\ge \sqrt{ab}

Variables: $a,b$ : non-negative real numbers.
Conditions: $a,b\ge 0$ ; equality iff $a=b$ .
Where used in JEE: Maximum-minimum problems, inequality proofs, expression bounds.

Equivalent AM-GM form for two numbersEssential

a+b\ge 2\sqrt{ab}

Variables: $a,b$ : non-negative real numbers.
Conditions: $a,b\ge 0$ ; equality iff $a=b$ .
Where used in JEE: Direct lower bound of sums and simplification in inequalities.

AM-GM inequality for n non-negative numbersEssential

\frac{x_1+x_2+\cdots+x_n}{n}\ge (x_1x_2\cdots x_n)^{1/n}

Variables: $x_1,x_2,\dots,x_n$ : non-negative real numbers.
Conditions: All $x_i\ge 0$ ; equality iff $x_1=x_2=\cdots=x_n$ .
Where used in JEE: General inequality problems, optimization, symmetric expressions.

Weighted AM-GM inequalityImportant

\frac{\alpha x+\beta y}{\alpha+\beta}\ge (x^{\alpha}y^{\beta})^{1/(\alpha+\beta)}

Variables: $x,y$ : non-negative reals, $\alpha,\beta$ : positive weights.
Conditions: $x,y\ge 0$ , $\alpha,\beta>0$ ; equality iff $x=y$ .
Where used in JEE: Weighted optimization and inequality transformations.

Three-variable AM-GMEssential

\frac{a+b+c}{3}\ge \sqrt[3]{abc}

Variables: $a,b,c$ : non-negative real numbers.
Conditions: $a,b,c\ge 0$ ; equality iff $a=b=c$ .
Where used in JEE: Standard 3-variable inequality questions.

Product bound from fixed sum by AM-GMImportant

x_1x_2\cdots x_n\le \left(\frac{x_1+x_2+\cdots+x_n}{n}\right)^n

Variables: $x_1,x_2,\dots,x_n$ : non-negative real numbers.
Conditions: All $x_i\ge 0$ ; equality iff all are equal.
Where used in JEE: Maximizing product for given sum.

Sum bound from fixed product by AM-GMImportant

x_1+x_2+\cdots+x_n\ge n(x_1x_2\cdots x_n)^{1/n}

Variables: $x_1,x_2,\dots,x_n$ : non-negative real numbers.
Conditions: All $x_i\ge 0$ ; equality iff all are equal.
Where used in JEE: Minimizing sum for given product.

Reciprocal form of AM-GM for two positive numbersImportant

\frac{1}{a}+\frac{1}{b}\ge \frac{4}{a+b}

Variables: $a,b$ : positive real numbers.
Conditions: $a,b>0$ ; equality iff $a=b$ .
Where used in JEE: Rational inequality simplification and bounds.

Square form of AM-GM for two real numbersImportant

a^2+b^2\ge 2ab

Variables: $a,b$ : real numbers.
Conditions: Equality iff $a=b$ .
Where used in JEE: Algebraic inequalities, deriving AM-GM-type bounds.

Positive sum-reciprocal inequalitySupplementary

(a+b)\left(\frac{1}{a}+\frac{1}{b}\right)\ge 4

Variables: $a,b$ : positive real numbers.
Conditions: $a,b>0$ ; equality iff $a=b$ .
Where used in JEE: Inequalities involving a variable and its reciprocal.

AM-GM equality conditionEssential

\text{AM}=\text{GM}\iff x_1=x_2=\cdots=x_n

Variables: $x_1,x_2,\dots,x_n$ : non-negative real numbers.
Conditions: Applicable in standard AM-GM inequality.
Where used in JEE: Determining equality case in optimization and bound problems.

Frequently asked questions

What are the important Sequence and Series formulas for JEE?

This page lists 45+ JEE-relevant Sequence and Series formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Sequence and Series formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Sequence and Series, covering Arithmetic and Geometric progressions, Insertion of arithmetic, geometric means between two given numbers, Relation between A.M and G.M.

How should I revise the Sequence and Series formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Sequence and Series MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Sequence and Series?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.