Math · Algebra

Sequence and Series formulas for JEE

Every Sequence and Series formula you need for JEE, grouped by concept.

FormulasRevision notes
25 formulas5 concepts
01Arithmetic Progression02Geometric Progression03Relation between AM and GM04AM-GM Inequality05Special Series Summation
01

Arithmetic Progression

3 formulas

Insertion of n A.M.s

d=b−an+1d = \frac{b-a}{n+1}

Common difference when inserting n arithmetic means between a and b.

applies whena and b are real numbers.
meansarithmeticjee-advanced

N-th term of A.P.

an=a+(n−1)da_n = a + (n-1)d

General term of an Arithmetic Progression.

applies whenValid for finite and infinite A.P.
progressionarithmeticjee-advanced

Sum of n terms of A.P.

Sn=n2[2a+(n−1)d]=n2(a+l)S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a+l)

Summation of the first n terms of an Arithmetic Progression.

applies whenn must be a natural number.
progressionarithmeticseriesjee-advanced
02

Geometric Progression

9 formulas

Insertion of n G.M.s

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

Common ratio when inserting n geometric means between positive numbers a and b.

applies whena and b are positive real numbers.
meansgeometric

Product of n G.M.s

G1G2…Gn=(ab)nG_1 G_2 \dots G_n = (\sqrt{ab})^n

The product of n geometric means inserted between two numbers equals the nth power of their single G.M.

applies whena and b are positive real numbers.
meansproductjee-advanced

N-th term of G.P.

an=arn−1a_n = a r^{n-1}

General term of a Geometric Progression.

applies whena is non-zero.
progressiongeometricsequence

G.P. Sum, Product, Reciprocal Relation

P2Rn=SnP^2 R^n = S^n

Relation between sum S, product P, and sum of reciprocals R of n terms of a G.P.

applies whenTerms must be non-zero.
progressiongeometricproperties

Sum of n terms of G.P.

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

Summation formula for a finite geometric series.

applies whenr≠1r \neq 1
progressiongeometricseries

Sum of G.P. (r=1)

Sn=naS_n = na

Sum of finite G.P. when the common ratio is 1.

applies whenr=1r = 1
progressiongeometricseries

Ratio of G.P. Sums

SnS2n−Sn=1rn\frac{S_n}{S_{2n} - S_n} = \frac{1}{r^n}

Ratio of the sum of first n terms to the sum of next n terms.

applies whenr≠1r \neq 1
progressiongeometricproperties

Sum of Infinite G.P.

S∞=a1−rS_\infty = \frac{a}{1-r}

Summation of a converging infinite geometric progression.

applies when∣r∣<1|r| < 1
progressiongeometricinfinitejee-advanced

GP sum

Sn=arn−1r−1S_n=a\frac{r^n-1}{r-1}

Sum of first n terms of a GP.

applies whenr\ne 1.
seriesgp
Reading a formula once isn't the same as recalling it in the exam. Rhovecs tracks which of these you've seen and brings them back on a forgetting schedule.See how it works
03

Relation between AM and GM

4 formulas

Algebraic Identity for roots

(a−b)2=(a+b)2−4ab(a-b)^2 = (a+b)^2 - 4ab

Identity often used to find numbers given their sum (from AM) and product (from GM).

identityalgebra

Relation between AM and GM

A=a+b2,G=abA = \frac{a+b}{2}, G = \sqrt{ab}

Arithmetic Mean (A) and Geometric Mean (G) of two numbers.

applies whena and b are positive real numbers.
meansinequalityrelation

Quadratic Equation from AM and GM

x2−2Ax+G2=0x^2 - 2Ax + G^2 = 0

Forming a quadratic equation when the A.M. and G.M. of its roots are given.

applies whenA and G are the AM and GM of the roots, respectively.
quadraticrootsmeans

AM-GM-HM Inequality

A.M.≥G.M.≥H.M.A.M. \geq G.M. \geq H.M.

Relationship between Arithmetic, Geometric, and Harmonic means.

applies whenStrictly for positive real numbers.
inequalitymeansjee-advanced
04

AM-GM Inequality

3 formulas

Relation between AM and GM

A=a+b2,G=abA = \frac{a+b}{2}, G = \sqrt{ab}

Arithmetic Mean (A) and Geometric Mean (G) of two numbers.

applies whena and b are positive real numbers.
meansinequalityrelation

AM-GM Inequality

A≥GA \geq G

The Arithmetic Mean is always greater than or equal to the Geometric Mean.

applies whenValid for positive real numbers. Equality holds when a = b.
inequalityoptimization

AM-GM-HM Inequality

A.M.≥G.M.≥H.M.A.M. \geq G.M. \geq H.M.

Relationship between Arithmetic, Geometric, and Harmonic means.

applies whenStrictly for positive real numbers.
inequalitymeansjee-advanced
05

Special Series Summation

6 formulas

Fibonacci Sequence

an=an−1+an−2a_n = a_{n-1} + a_{n-2}

Recurrence relation generating the Fibonacci sequence.

applies whenn>2,a1=1,a2=1n > 2, a_1 = 1, a_2 = 1
sequencerecurrence

Sum of Infinite A.G.P.

S∞=a1−r+dr(1−r)2S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}

Sum of a converging infinite Arithmetico-Geometric Progression.

applies when∣r∣<1|r| < 1
seriesagpinfinitejee-advanced

Sum of n terms of A.G.P.

Sn=a1−r+dr(1−rn−1)(1−r)2−[a+(n−1)d]rn1−rS_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{[a+(n-1)d]r^n}{1-r}

Sum of a finite Arithmetico-Geometric Progression.

applies whenr≠1r \neq 1
seriesagpjee-advanced

Sum of first n natural numbers

∑k=1nk=n(n+1)2\sum_{k=1}^n k = \frac{n(n+1)}{2}

Summation of consecutive integers starting from 1.

applies whenn is a natural number.
seriessummationjee-advanced

Sum of squares of first n natural numbers

∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}

Summation of the squares of consecutive integers.

applies whenn is a natural number.
seriessummationjee-advanced

Sum of cubes of first n natural numbers

∑k=1nk3=[n(n+1)2]2\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2

Summation of the cubes of consecutive integers.

applies whenn is a natural number.
seriessummationjee-advanced
Other chapters
MathSets34 formulasMathRelations and Functions18 formulasMathTrigonometric Functions56 formulasMathComplex Numbers and Quadratic Equations36 formulasMathPermutations and Combinations21 formulasMathBinomial Theorem24 formulasMathStraight Lines27 formulasMathConic Sections35 formulasMathIntroduction to Three-dimensional Geometry9 formulasMathLimits and Derivatives32 formulasMathStatistics25 formulasMathProbability17 formulasMathRelations and Functions14 formulasMathInverse Trigonometric Functions33 formulasMathMatrices44 formulasMathDeterminants23 formulasMathContinuity and Differentiability30 formulasMathApplications of Derivatives26 formulasMathIntegrals47 formulasMathApplications of the Integrals14 formulasMathDifferential Equations23 formulasMathVectors29 formulasMathThree-dimensional Geometry32 formulasMathProbability15 formulas

Rhovecs schedules these formulas back to you right before you’d forget them — and picks the next concept to practise. We decide, you execute.

Get started