Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 10

HARD

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IMPORTANT

Earn 100

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. Total $N$ number of toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles. Then find the sum of all digits of $N .$

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Important Questions on Sequences and Series

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 11

A B C

is a right-angled triangle in which

∠ B = 90 °

and

B C = a .

If

n

points

L_{1}, L_{2}, \dots \dots, L_{n}

on

A B

is divided in

n + 1

equal parts and

L_{1} M_{1}, L_{2} M_{2}, \dots \dots L_{n} M_{n}

are line segments parallel to

B C

and

M_{1}, M_{2}, \dots \dots \dots . M_{n}

are on

A C,

then the sum of the lengths of

L_{1}^{} M_{1}, L_{2} M_{2}, \dots . . L_{n} M_{n}

is

MEDIUM

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 12

If

\tan \frac{2 π}{18}, x

and

\tan \frac{7 π}{18}

are in A.P. and

\tan \frac{2 π}{18}, y

and

\tan \frac{5 π}{18}

are in A.P., then the value of

\frac{x}{y}

will be

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 13

Given the sequence of numbers

x_{1}, x_{2}, x_{3}, x_{4}, \dots, x_{2005}

which satisfy

\frac{x_{1}}{x_{1} + 1} = \frac{x_{2}}{x_{2} + 3} = \frac{x_{3}}{x_{3} + 5} = \dots = \frac{x_{2005}}{x_{2005} + 4009}

, the nature of the sequence is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 14

If

a_{1}, a_{2}, a_{3}, \dots, a_{87}, a_{88}, a_{89}

are the arithmetic means between

1

and

89,

then

\sum_{r = 1}^{89} \log (\tan (a_{r}) °)

is equal to

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 15

The sum of all the elements in the set

{n \in {1, 2, \dots \dots, 100} ∣

H.C.F. of

n

and

2040

is

1

} is equal to __________.

EASY

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 16

If the first, second and last term of an arithmetic series are

a, b, c

respectively, them the number of terms is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 17

If

a, b, c

are in arithmetic progression, then the value of

(a + 2 b - c) (2 b + c - a) (a + 2 b + c)

is

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 25 - Sequences and Series>Level 2>Q 18

Let

A_{1}, A_{2}, A_{3}, \dots . .

be squares such that for each

n ⩾ 1,

the length of the side of

A_{n}

equals the length of diagonal of

A_{n + 1}

. If the length of

A_{1}

is

12 cm

, then the smallest value of

n

for which area of

A_{n}

is less than one, is