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The number of terms common between the series $1 + 2 + 4 + 8 . . .$ to $100$ terms and $1 + 4 + 7 + 10 + . .$ to $100$ terms is

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Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence 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>Sequence 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

EASY

Mathematics>Algebra>Sequence and Series>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>Sequence 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>Sequence 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>Sequence 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 .

HARD

Mathematics>Algebra>Sequence 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

MEDIUM

Mathematics>Algebra>Sequence and Series>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 ____________.

MEDIUM

Mathematics>Algebra>Sequence 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 :

MEDIUM

Mathematics>Algebra>Sequence 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

HARD

Mathematics>Algebra>Sequence and Series>Arithmetic Progression

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

and

^{n} C_{6}

are in A.P., then

n

can be

MEDIUM

Mathematics>Algebra>Sequence and Series>Arithmetic Progression

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

7

yield

2

5

as remainder is

MEDIUM

Mathematics>Algebra>Sequence and Series>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>Sequence 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

MEDIUM

Mathematics>Algebra>Sequence and Series>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 :

HARD

Mathematics>Algebra>Sequence and Series>Arithmetic Progression

Let

A P (a; d)

denote the set of all the terms of an infinite arithmetic progression with first term

a

and common difference

d > 0 .

A P (1; 3) \cap A P (2; 5) \cap A P (3; 7) = A P (a; d)

then

a + d

equals ____

MEDIUM

Mathematics>Algebra>Sequence and Series>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>Sequence 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

MEDIUM

Mathematics>Algebra>Sequence 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.

HARD

Mathematics>Algebra>Sequence and Series>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___.

The number of terms common between the series 1+2+4+8 ... to 100 terms and 1+4+7+10+ .. to 100 terms is

Important Questions on Sequence and Series

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The number of terms common between the series $1 + 2 + 4 + 8 . . .$ to $100$ terms and $1 + 4 + 7 + 10 + . .$ to $100$ terms is