Core Syllabus Module

Sequence and Series

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Generate random timed arrays filtered strictly by Sequence and Series weightage mappings.

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Main 2022 (28 July Shift 1)

LEVELJEE Advanced

Let $x_{1}, x_{2}, x_{3}, \dots, x_{20}$ be in geometric progression with $x_{1} = 3$ and the common ratio $\frac{1}{2}$ . A new data is constructed replacing each $x_{i}$ by $(x_{i} - i)^{2}$ . If $\overset{x}{ˉ}$ is the mean of new data, then the greatest integer less than or equal to $\overset{x}{ˉ}$ is ________.

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Advanced

Let $a = 1 + \frac{^{2} C _{2}}{3 !} + \frac{^{3} C _{2}}{4 !} + \frac{^{4} C _{2}}{5 !} + \dots$ , $b = 1 + \frac{^{1} C _{0} + ^{1} C _{1}}{1 !} + \frac{^{2} C _{0} + ^{2} C _{1} + ^{2} C _{2}}{2 !} + \frac{^{3} C _{0} + ^{3} C _{1} + ^{3} C _{2} + ^{3} C _{3}}{3 !} + \dots$ . Then $\frac{2 b}{a ^{2}}$ is equal to

JEE Main 2024 (05 Apr Shift 2)

LEVELJEE Advanced

If $1 + \frac{3 - 2}{2 3} + \frac{5 - 2 6}{18} + \frac{9 3 - 11 2}{36 3} + \frac{49 - 20 6}{180} + \dots upto \infty = 2 + (\frac{b}{a} + 1) lo g_{e} (\frac{a}{b})$ , where $a$ and $b$ are integers with $gcd (a, b) = 1$ , then $11 a + 18 b$ is equal to

JEE Main 2003

LEVELJEE Advanced

Let $R_{1}$ and $R_{2}$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_{1}, R, R_{2}$ are in

JEE Main 2023 (24 January Shift 1)

LEVELJEE Advanced

For three positive integers $p, q, r$ , $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and $r = pq + 1$ such that $3, 3 lo g_{y} x, 3 lo g_{z} y, 7 lo g_{x} z$ are in A.P. with common difference $\frac{1}{2}$ . Then $r - p - q$ is equal to

JEE Main 2022 (29 July Shift 2)

LEVELJEE Advanced

Let ${a_{n}}_{n = 0}^{\infty}$ be a sequence such that $a_{0} = a_{1} = 0$ and $a_{n + 2} = 3 a_{n + 1} - 2 a_{n} + 1, \forall n \geq 0$ . Then $a_{25} a_{23} - 2 a_{25} a_{22} - 2 a_{23} a_{24} + 4 a_{22} a_{24}$ is equal to:

JEE Main 2019 (08 April Shift 2)

LEVELJEE Advanced

If three distinct numbers a,b,c are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

(A)

d, e, f

are in A.P.

(B)

d / a, e / b, f / c

are in G.P.

(C)

d / a, e / b, f / c

are in A.P.

(D)

d, e, f

are in G.P.

JEE Advanced 2006

LEVELJEE Advanced

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

JEE Advanced 2019

LEVELJEE Advanced

Let $α$ and $β$ be the roots of $x^{2} - x - 1 = 0$ , with $α > β$ . For all positive integers $n$ , define $a_{n} = \frac{α ^{n} - β ^{n}}{α - β}, n \geq 1$ , $b_{1} = 1$ and $b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2$ . Then which of the following options is/are correct?

* Multiple Correct Options

(A)

a_{1} + a_{2} + a_{3} + \dots + a_{n} = a_{n + 2} - 1

for all

n \geq 1

(B)

\sum_{n = 1}^{\infty} \frac{a _{n}}{1 0 ^{n}} = \frac{10}{89}

(C)

\sum_{n = 1}^{\infty} \frac{b _{n}}{1 0 ^{n}} = \frac{8}{89}

(D)

b_{n} = α^{n} + β^{n}

for all

n \geq 1

JEE Main 2022 (28 July Shift 1)

LEVELJEE Main

If the minimum value of $f (x) = \frac{5 x ^{2}}{2} + \frac{α}{x ^{5}}, x > 0$ , is 14, then the value of $α$ is equal to :

JEE Main 2023 (31 January Shift 1)

LEVELJEE Main

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

(A)

(B)

\frac{9}{2}

(C)

(D)

JEE Main 2026 (24 January Shift 2)

LEVELJEE Main

$(\frac{1}{3} + \frac{4}{7}) + (\frac{1}{3 ^{2}} + \frac{1}{3} \times \frac{4}{7} + \frac{4 ^{2}}{7 ^{2}}) + (\frac{1}{3 ^{3}} + \frac{1}{3 ^{2}} \times \frac{4}{7} + \frac{1}{3} \times \frac{4 ^{2}}{7 ^{2}} + \frac{4 ^{3}}{7 ^{3}}) + \dots$ upto infinite terms, is equal to

Comprehension Passage

Let

V_{r}

denote the sum of first

r

terms of an arithmetic progression (A.P.) whose first term is

r

and the common difference is

(2 r - 1)

. Let

T_{r} = V_{r + 1} - V_{r} - 2

and

Q_{r} = T_{r + 1} - T_{r}

for

r = 1, 2, \dots

Question 1:

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

(A)

\frac{1}{12} n (n + 1) (3 n^{2} - n + 1)

(B)

\frac{1}{12} n (n + 1) (3 n^{2} + n + 2)

(C)

\frac{1}{3} n (2 n^{2} - n + 1)

(D)

\frac{1}{3} (2 n^{3} - 2 n + 3)

Question 2:

$T_{r}$ is always

Which one of the following is a correct statement ?

(A)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 5

(B)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 6

(C)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 11

(D)

Q_{1} = Q_{2} = Q_{3} = \dots

JEE Advanced 1982

LEVELJEE Main

$mn$ squares of equal size are arranged to from a rectangle of dimension $m$ by $n$ , where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.

JEE Advanced 1991

LEVELJEE Main

Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between $p$ and $(\frac{n + 1}{n - 1})^{2} p$ .

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

Let ${a_{n}}_{n = 0}^{\infty}$ be a sequence such that $a_{0} = a_{1} = 0$ and $a_{n + 2} = 2 a_{n + 1} - a_{n} + 1$ for all $n \geq 0$ . Then, $\sum_{n = 2}^{\infty} \frac{a _{n}}{7 ^{n}}$ is equal to

(A)

\frac{6}{343}

(B)

\frac{7}{216}

(C)

\frac{8}{343}

(D)

\frac{49}{216}

JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

Let $a_{1} = b_{1} = 1$ and $a_{n} = a_{n - 1} + (n - 1), b_{n} = b_{n - 1} + a_{n - 1}, \forall n \geq 2$ . If $S = \sum_{n = 1}^{10} \frac{b _{n}}{2 ^{n}}$ and $T = \sum_{n = 1}^{8} \frac{n}{2 ^{n - 1}}$ , then $2^{7} (2 S - T)$ is equal to

JEE Main 2022 (28 July Shift 1)

LEVELJEE Main

Consider the sequence $a_{1}, a_{2}, a_{3}, \dots$ such that $a_{1} = 1, a_{2} = 2$ and $a_{n + 2} = \frac{2}{a _{n + 1}} + a_{n}$ for $n = 1, 2, 3, \dots$ . If $(\frac{a _{1} + \frac{1}{a _{2}}}{a _{3}}) \cdot (\frac{a _{2} + \frac{1}{a _{3}}}{a _{4}}) \cdot (\frac{a _{3} + \frac{1}{a _{4}}}{a _{5}}) \dots (\frac{a _{30} + \frac{1}{a _{31}}}{a _{32}}) = 2^{α} (31 61)$ , then $α$ is equal to :

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1, 6, 11, \dots$ , and $Y$ be the set consisting of the first 2018 terms of arithmetic progression $9, 16, 23, \dots$ . Then, the number of elements in the set $X \cup Y$ is ________.

JEE Main 2021 (26 February Shift 2)

LEVELJEE Main

The sum of the series $\sum_{n = 1}^{\infty} \frac{n ^{2} + 6 n + 10}{( 2 n + 1 )!}$ is equal to :

(A)

\frac{41}{8} e + \frac{19}{8} e^{- 1} - 10

(B)

- \frac{41}{8} e + \frac{19}{8} e^{- 1} - 10

(C)

\frac{41}{8} e - \frac{19}{8} e^{- 1} - 10

(D)

\frac{41}{8} e + \frac{19}{8} e^{- 1} + 10

Micro-Topic Strategy Matrix

Arithmetic Progression (A.P.)

Geometric Progression (G.P.)

Harmonic Progression (H.P.)

Arithmetic-Geometric Progression (A.G.P.)

Relation Between A.M., G.M., and H.M.

Sum of Special Series

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=1+3!2C2​​+4!3C2​​+5!4C2​​+…, b=1+1!1C0​+1C1​​+2!2C0​+2C1​+2C2​​+3!3C0​+3C1​+3C2​+3C3​​+…. Then a22b​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R1​ and R2​ respectively be the maximum ranges up and down an inclined plane and R be the maximum range on the horizontal plane. Then R1​,R,R2​ are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For three positive integers p,q,r, xpq2=yqr=zp2r and r=pq+1 such that 3,3logy​x,3logz​y,7logx​z are in A.P. with common difference 21​. Then r−p−q is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If three distinct numbers a,b,c are in G.P. and the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which one of the following statements is correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=43​−(43​)2+(43​)3+⋯+(−1)n−1(43​)n and bn​=1−an​, then find the least natural number n0​ such that bn​>an​ for all n≥n0​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the minimum value of f(x)=25x2​+x5α​,x>0, is 14, then the value of α is equal to :

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (31​+74​)+(321​+31​×74​+7242​)+(331​+321​×74​+31​×7242​+7343​)+… upto infinite terms, is equal to

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum V1​+V2​+⋯+Vn​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr​ is always

Which one of the following is a correct statement ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let {an​}n=0∞​ be a sequence such that a0​=a1​=0 and an+2​=2an+1​−an​+1 for all n≥0. Then, ∑n=2∞​7nan​​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​=b1​=1 and an​=an−1​+(n−1),bn​=bn−1​+an−1​,∀n≥2. If S=∑n=110​2nbn​​ and T=∑n=18​2n−1n​, then 27(2S−T) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the series ∑n=1∞​(2n+1)!n2+6n+10​ is equal to :

Let $R_{1}$ and $R_{2}$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_{1}, R, R_{2}$ are in

For three positive integers $p, q, r$ , $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and $r = pq + 1$ such that $3, 3 lo g_{y} x, 3 lo g_{z} y, 7 lo g_{x} z$ are in A.P. with common difference $\frac{1}{2}$ . Then $r - p - q$ is equal to

If three distinct numbers a,b,c are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

If the minimum value of $f (x) = \frac{5 x ^{2}}{2} + \frac{α}{x ^{5}}, x > 0$ , is 14, then the value of $α$ is equal to :

$(\frac{1}{3} + \frac{4}{7}) + (\frac{1}{3 ^{2}} + \frac{1}{3} \times \frac{4}{7} + \frac{4 ^{2}}{7 ^{2}}) + (\frac{1}{3 ^{3}} + \frac{1}{3 ^{2}} \times \frac{4}{7} + \frac{1}{3} \times \frac{4 ^{2}}{7 ^{2}} + \frac{4 ^{3}}{7 ^{3}}) + \dots$ upto infinite terms, is equal to

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

$T_{r}$ is always

Let ${a_{n}}_{n = 0}^{\infty}$ be a sequence such that $a_{0} = a_{1} = 0$ and $a_{n + 2} = 2 a_{n + 1} - a_{n} + 1$ for all $n \geq 0$ . Then, $\sum_{n = 2}^{\infty} \frac{a _{n}}{7 ^{n}}$ is equal to

Let $a_{1} = b_{1} = 1$ and $a_{n} = a_{n - 1} + (n - 1), b_{n} = b_{n - 1} + a_{n - 1}, \forall n \geq 2$ . If $S = \sum_{n = 1}^{10} \frac{b _{n}}{2 ^{n}}$ and $T = \sum_{n = 1}^{8} \frac{n}{2 ^{n - 1}}$ , then $2^{7} (2 S - T)$ is equal to

The sum of the series $\sum_{n = 1}^{\infty} \frac{n ^{2} + 6 n + 10}{( 2 n + 1 )!}$ is equal to :