For distinct positive numbers a , b and c , if a 2 , b 2 and c 2 are in AP , then which one of the following triplets is also in AP ?

1 b + 2 c , 1 c + 2 a , 1 a + 2 b

MEDIUM

JEE Main

IMPORTANT

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Consider a sequence of real numbers $\{a_{n}\}$ defined by $a_{1} = 1$ and $a_{n + 1} = \frac{a_{n}}{1 + n a_{n}}$ for $n \geq 1$ .

Find the value of $\frac{1}{a_{2005}} - 2009000$

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Important Questions on Sequences and Series

MEDIUM

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 42

Let

m

be the smallest odd positive integer for which

1 + 2 + \dots \dots + m

is a square of an integer, and let

n

be the smallest even positive integer for which

1 + 2 + \dots \dots + n

is a square of an integer. What is the value of

m + n ?

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 1

3 x^{2} - 2 a x + (a^{2} + 2 b^{2} + 2 c^{2}) = 2 (a b + b c)

, then

a, b, c

can be in

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 2

How many four-digit integers

a b c d,

with

a \neq 0,

have the property that the three two-digit integers

a b < b c < c d

form an increasing arithmetic sequence? One such number is

4692,

where

a = 4, b = 6, c = 9,

and

d = 2 .

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 3

A

B, C

are three piles of rocks. The mean weight of the rocks in

A

40 kg,

the mean weight of the rocks in

B

50 kg,

the mean weight of the rocks in the combined piles

A

and

B

43 kg,

and the mean weight of the rocks in the combined piles

A

and

C

44 kg .

What is the greatest possible integer value for the mean in

kg

of the rocks in the combined piles

B

and

C ?

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 4

The line $x + y = 1$ meets $x$ -axis at $A$ and $y$ -axis at $B,$ $P$ is the mid-point of $A B; P_{1}$ is the foot of the perpendicular from $P$ to $O A; M_{1}$ is that of $P_{1}$ on $O P; P_{2}$ is that of $M_{1}$ on $O A; M_{2}$ is that of $P_{2}$ on $O P; P_{3}$ is that of $M_{2}$ on $O A;$ and so on.

If $P_{n}$ denotes the $n^{th}$ foot of the perpendicular on $O A;$ then $O P_{n}$ is equal to

Question Image

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 5

Let

f : N \to N

be such that

f (1) = 2

and

f (x + y) = f (x) f (y)

for all natural numbers

x

and

y

. If

\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1),

then

a

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 6

Let

a, b, c, d

and

e

be real numbers such that

a > b > 0

. If

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

are in

AP

b, c, d

are in

G P

and

c

d, e

are in

A P

, then

\frac{a b^{2}}{(2 a - b)^{2}}

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 3>Q 7

For each integer

n \geq 4,

let

a_{n}

denote the base-

n

number

0 . {\bar{133}}_{n} .

The product

a_{4} a_{5} \dots \dots \dots . a_{99}

can be expressed as

\frac{m}{n!}

, where

m

and

n

are positive integers and

n

is as small as possible. What is the value of sum of the digits of

m

Consider a sequence of real numbers an defined by a1=1 and an+1=an1+nan for n≥1. Find the value of 1a2005-2009000

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Consider a sequence of real numbers $\{a_{n}\}$ defined by $a_{1} = 1$ and $a_{n + 1} = \frac{a_{n}}{1 + n a_{n}}$ for $n \geq 1$ .

Find the value of $\frac{1}{a_{2005}} - 2009000$