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The sum of two numbers is $\frac{13}{6}$ . An even number of arithmetic means are inserted between them and their sum exceeds their number by $1$ . Find the number of means inserted.

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Important Questions on Sequences and Series

HARD

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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>A.M. and Useful Results in 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

EASY

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

In an arithmetic progression, if the

k^{t h}

term is

5 k + 1

,

then the sum of first

100

terms is

HARD

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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>A.M. and Useful Results in 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>A.M. and Useful Results in 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>A.M. and Useful Results in 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

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

The number of terms common to the two A.P.’s

3, 7, 11, \dots, 407

and

2, 9, 16, \dots, 709

is ____________.

HARD

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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 :

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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>A.M. and Useful Results in 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>A.M. and Useful Results in Arithmetic Progression

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

and

^{n} C_{6}

are in A.P., then

n

can be

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

Let

a_{1}, a_{2}, a_{3}, \dots \dots, a_{49}

be in

A . P .

such that

Σ_{k = 0}^{12} a_{4 k + 1} = 416

and

a_{9} + a_{43} = 66 .

a_{1}^{2} + a_{2}^{2} + \dots + a_{17}^{2} = 140 m,

then

m

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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:

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

Let

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

,be in A.P. If

a_{3} + a_{7} + a_{11} + a_{15} = 72

, then the sum of its first

17

terms is equal to :

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

If the sum and product of the first three terms in an

A . P .

are

33

and

1155

, respectively, then a value of its

11^{t h}

term is:

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

Let the sum of the first

n

terms of a non-constant

A . P ., a_{1}, a_{2}, a_{3}, . . . ., a_{n}

50 n + \frac{n (n - 7)}{2} A,

where

A

is a constant. If

d

is the common difference of this

A . P .

, then the ordered pair

(d, a_{50})

is equal to

HARD

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in 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>A.M. and Useful Results in Arithmetic Progression

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 the arithmetic progression

9,16,23, \dots

. Then, the number of elements in the set

X \cup Y

is___.

MEDIUM

Mathematics>Algebra>Sequences and Series>A.M. and Useful Results in Arithmetic Progression

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

7

yield

2

5

as remainder is

The sum of two numbers is 136. An even number of arithmetic means are inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

Important Questions on Sequences and Series

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The sum of two numbers is $\frac{13}{6}$ . An even number of arithmetic means are inserted between them and their sum exceeds their number by $1$ . Find the number of means inserted.