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If tan(A + B), tan B and tan(B + C) are in AP. Then, which of the following options are correct ?

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Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Three positive numbers form an increasing

G . P .

If the middle term in this

G . P .

is doubled, the new numbers are in

A . P .

Then the common ratio of the

G . P .

is :

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

^{n} C_{4},^{n} C_{5}

and

^{n} C_{6}

are in A.P., then

n

can be

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let the sum of the first three terms of an A.P. be

39

and the sum of its last four terms be

178

. If the first term of this A.P. is

10,

then the median of the A.P. is :

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

For any three positive real numbers

a, b

and

c

. If

9 (25 a^{2} + b^{2}) + 25 (c^{2} - 3 a c) = 15 b (3 a + c) .

Then

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

a_{1}, a_{2}, \dots, a_{30}

be an A.P.,

S = \sum_{i = 1}^{30} a_{i}

and

T = \sum_{i = 1}^{15} a_{(2 i - 1)} .

a_{5} = 27

and

S - 2 T = 75,

then

a_{10}

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

\frac{1}{x_{1}}, \frac{1}{x_{2}}, \dots, \frac{1}{x_{n}} (x_{i} \neq 0

for

i = 1, 2, \dots ., n)

be in A.P. such that

x_{1} = 4

and

x_{21} = 20

.

n

is the least positive integer for which

x_{n} > 50

, then

\sum_{i = 1}^{n} (\frac{1}{x_{i}})

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

x, y, z

are positive numbers in

A . P .

and

\tan^{- 1} x

\tan^{- 1} y

and

\tan^{- 1} z

are also in

A . P .

, then which of the following is correct.

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

a^{1 / x} = b^{1 / y} = c^{1 / z}

and

a, b, c

are in geometrical progression, then

x, y, z

are in

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

If $m$ is the $A . M .$ of two distinct real numbers $I$ and $n$ $(I, n > 1)$ and $G_{1}, G_{2} and G_{3}$ are three geometric means between $I and n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

The sum of all two digit positive numbers which when divided by

7

yield

2

5

as remainder is

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

a, b, c, d, e

be natural numbers in an arithmetic progression such that

a + b + c + d + e

is the cube of an integer and

b + c + d

is a square of an integer. The least possible value of the number of digits of

c

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

S_{n} = Σ_{k = 1}^{4 n} {(- 1)}^{\frac{k (k + 1)}{2}} k^{2}

. Then

S_{n}

can take value(s)

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is

170

. If there are at least

6

houses in that row and

a

is the number of the sixth house, then

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

α and β

be the roots of equation

p x^{2} + q x + r = 0

p \neq 0

. If

p, q, r

are in A.P. and

\frac{1}{α} + \frac{1}{β} = 4

, then the value of

|α - β|

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

If the sum of the first

n

terms of the series

\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \dots

435 \sqrt{3},

then

n

equals:

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than

200

and less than

220 .

If the second term in it is

12

, then its

4^{th}

term is :

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

The number of terms in an

A . P .

is even, the sum of the odd terms in it is

24

and that the even terms is

30 .

If the last term exceeds the first term by

10 \frac{1}{2},

then the number of terms in the

A . P .

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

a, b

and

c

be the

7^{t h}, 11^{t h}

and

13^{t h}

terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then

\frac{a}{c}

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

The sum of the first

10

terms of the series

9 + 99 + 999 + . \dots .

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progression

Let

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

be an arithmetic progression with

a_{1} = 7

and common difference

8

. Let

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

be such that

T_{1} = 3

and

T_{n + 1} - T_{n} = a_{n}

n \geq 1

. Then, which of the following is/are TRUE?

If tan(A + B), tan B and tan(B + C) are in AP. Then, which of the following options are correct ?

Important Questions on Sequences and Series

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