Let S n n ≥ 1 be a sequence of sets defined by S 1 = 0 , S 2 = 3 2 , 5 2 , S 3 = 8 3 , 11 3 , 14 3 , S 4 = 15 4 , 19 4 , 23 4 , 27 4 , … . , then

third element in S 20 is 431 20

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Let $S_{n} (n \geq 1)$ be a sequence of sets defined by $S_{1} = \{0\}, S_{2} = \{\frac{3}{2}, \frac{5}{2}\}, S_{3} = \{\frac{8}{3}, \frac{11}{3}, \frac{14}{3}\}, S_{4} = \{\frac{15}{4}, \frac{19}{4}, \frac{23}{4}, \frac{27}{4}\}, \dots .$ , then

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Important Questions on Sequences and Series

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Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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

HARD

Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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|

HARD

Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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___.

HARD

Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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 :

MEDIUM

Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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>Series of Natural Numbers and other Miscellaneous Series

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

and

^{n} C_{6}

are in A.P., then

n

can be

MEDIUM

Mathematics>Algebra>Sequences and Series>Series of Natural Numbers and other Miscellaneous Series

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

7

yield

2

5

as remainder is

Let Snn≥1 be a sequence of sets defined by S1=0,S2=32,52,S3=83,113,143,S4=154,194,234,274,…., then

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Let $S_{n} (n \geq 1)$ be a sequence of sets defined by $S_{1} = \{0\}, S_{2} = \{\frac{3}{2}, \frac{5}{2}\}, S_{3} = \{\frac{8}{3}, \frac{11}{3}, \frac{14}{3}\}, S_{4} = \{\frac{15}{4}, \frac{19}{4}, \frac{23}{4}, \frac{27}{4}\}, \dots .$ , then