Sequence & Series JEE 2026 – Easy Complete notes

Sequence & Series JEE 2026 – Easy Complete notes is one of the most important and scoring topics in JEE Main 2026. Questions frequently asked from A.P, G.P and series sum.

Sequence and series

Sequence : A set of numbers arranged in a definite order according to some definite rule is called a sequence

Examples : 1, 3 , 5, 7, 9……………………. is a sequence

Series : The sum of terms of a sequence is called a series.

Examples : If a₁, a₂, a₃ ………….. is a sequence, then the expression a₁ + a₂ + a₃ +………… is a series.

Finite and infinite series : A series is called a finite series If it has Finite number of terms, otherwise it is called infinite series

Example : 2 + 4 + 6 + 8 + ………………….. + 48 – finite series

Example : 1 + 4 + 7 + 10 + …………….. – infinite series

Progressions : Sequence which follow a specific pattern are called progressions

(i) Arithmetic Progression

(ii) Geometric Progression

(iii) Harmonic Progression

(iv) Arithmetic – Geometric Progression

Arithmetic Progression : A sequence is called an Arithmetic Progression, if the difference between any two consecutive terms is same

A.P – a, a + d, a + 2d, a + 3d, + …………….. + a + [n – 1]d

where

a is the first term

d is the common difference

General term of an A.P :

Let ‘a’ be the first term and d be the common difference of an A.P, then its general term (or) n^th term is given by

• a_n = a + [n – 1]d
• a_n = T_n

m^th term from end of an A.P :

If a is the first term and d is common difference of an A.P having ‘n’ terms, then the m^th term from the end is [ n – m + 1]^th term

T_m^th = [ n – m + 1]

We can also find the m^th term from the end as folllows

T_m = l – (m – 1)d

where l is last term

Some Shortcuts :

• If p^th term of an A.P is ‘q’ and q^th term is ‘P’ then T_{p + q} = 0
• If m^th term of an A.P is ‘n’ and n^th term is ‘m’ then p^th term is m + n – p
• For an A.P If S_p = q and S_q = p then S_{p + q} = -( p + q)

Sum of n terms of an A.P

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

• If the sum of n terms of a sequence S_n is given then we can find the n^th term of the sequence as T_n = S_n – S_{n – 1}

Note :

If S_n = pn² + qn the sequence is A.p with d = 2p

Properties of an A.P :

• If a, b , c are in A.P then 2b = a + c
• Sum of the terms equidistant from beginning and end is same for a_n A.P a₁, a₂, a₃, a₄, a₅……….. a_n we have a₁ + a_n = a₂ + a_{n – 1} = a₃ + a_{n – 2} = …………

If a₁, a₂, a₃, a₄, a₅……….. a_n are in A.P the

• a_n, a_{n – 1} ………….. a₂, a₁ are also in A.P
• a₁$\pm$k, a₂$\pm$k, a₃$\pm$k, a₄$\pm$k ……………….. a_n$\pm$k are also in A.P
• Ka₁, Ka₂, Ka₃ ………………………. Ka_n are also in A.P
• $\frac{a_1}{K}$, $\frac{a_2}{K}$, $\frac{a_3}{K}$ ………………………… $\frac{a_n}{K}$ are also in A.P

If a₁, a₂, a₃ …………. a_n are in A.P and b₁, b₂, b₃ ………………… b_n are also in A.P

• a₁$\pm$b₁, a₂$\pm$b₂, a₃$\pm$b₃ ……………… a_n$\pm$b_n are in A.P
• a₁b₁, a₂b₂, a₃b₃ ……………… and $\frac{a_1}{b_1}$, $\frac{a_2}{b_2}$, $\frac{a_3}{b_3}$ …………….. are not in A.P

If the terms of A.P are chosen at regular intervals then they form an A.P

Example : 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 , 21……………

1, 7, 13, 19 are in A.P

Common terms of two A.P :

The n^th Common term of two arithmetic series is 1^st common term + [ n – 1 ] [ LCM of common difference of 1^st series and 2^nd series ]

Selection of terms in A.P :

Arithmetic Mean :

• a, b are in GM $\rightarrow$ $\frac{a + b }{2}$
• a, b , c are in GM $\rightarrow$ $\frac{a + b + c }{3}$
• a₁, a₂, a₃, ………………… a_n $\rightarrow$ $\frac{a_1 + a_2 +a_3 + a_4 + ……………. +a_n}{n}$

Geometric Progressions :

• a, ar, ar² ……………………..
• A sequence is called an geometric progression, if the ratio of any two consecutive terms is same
• G.P is of the form a, ar, ar², ar³,……………………… ar^{n – 1} where a – First term , r – common ratio
• a₁, a₂, a₃ ……………… a_n are in G.P where r = $\frac{a_2}{a_1}$ = $\frac{a_3}{a_2}$ = ………
• If a, b , c are in GP $\rightarrow$ b² = ac
• No term of GP can be ‘0’
• T_n = ar^{n – 1}

n^th term from end of a G.P

The n^th term from the end of a finite GP consisting of m terms is T_{m – n + 1} = ar^{m – n}

[ or ]

a₁, a₂, a₃ ……………… a_n are in G.P

Reverse the G.P $\rightarrow$ a_n , a_{n – 1}, a_{n – 2}, ………………….. a₃, a₂, a₁

Common ratio – $\frac{1}{r}$

T_n = a_n($\frac{1}{r}$)^{n – 1}

Sum of n terms of a GP :

S_n = $\frac{a( r^n – 1)}{r – 1}$ r > 1

S_$\infin$ = $\frac{a}{1 – r}$ |r|< 1

Geometric Mean :

• a, b are in GM $\rightarrow$ $\sqrt{ab}$
• a, b, c are in GM $\rightarrow$ (abc)^{$\frac{1}{3}$}
• a₁,a₂, a₃ ……………….. a_n are in GM $\rightarrow$ (a₁, a₂, a₃, a₄ ……………………. a_n)^{$\frac{1}{n}$}

Relation b/w AM and GM :

If a, b > 0

AM$\ge$GM

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

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

If a, b, c > 0

AM $\ge$ GM

$\frac{ a+b + c}{3}$ $\ge$ (abc)^{$\frac{1}{3}$}

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