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The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Find whether the resulting sum is square of an integer or not?

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Important Questions on Sequence and Series

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

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

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

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>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

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

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

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>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 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_{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:

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 :

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

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

Let

a_{1}, a_{2}, a_{3} . . .

be an

A . P .

with

a_{6} = 2 .

Then, the common difference of this

A . P .,

which maximise the product

a_{1} \cdot a_{4} \cdot a_{5},

is :

HARD

Mathematics>Algebra>Sequence and Series>Arithmetic Progression

The sum of all natural numbers

n

such that

100 < n < 200

and

H . C . F .

(91, n) > 1

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

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

7

yield

2

5

as remainder is

EASY

Mathematics>Algebra>Sequence and Series>Arithmetic Progression

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

and

\frac{1}{h_{1}}, \frac{1}{h_{2}}, \dots . ., \frac{1}{h_{n}}

are two A.P.s such that

x_{3} = h_{2} = 8 & x_{8} = h_{7} = 20

, then

x_{5} \cdot h_{10}

is equal to

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Find whether the resulting sum is square of an integer or not?

Important Questions on Sequence and Series

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