HARD

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Find the sum of the following series:

$(2^{2} + 4^{2} + 6^{2} + . . . . . . . . . + 20^{2}) = ?$

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Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If the sum of the first

15

terms of the series

{(\frac{3}{4})}^{3} + {(1 \frac{1}{2})}^{3} + {(2 \frac{1}{4})}^{3} + 3^{3} + {(3 \frac{3}{4})}_{}^{3} + \dots

is equal to

225 K

, then

K

is equal to :

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

\sum_{n = 1}^{5} \frac{1}{n (n + 1) (n + 2) (n + 3)} = \frac{k}{3},

then

k

is equal to:

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

ω

, is an imaginary cube root of

1

, then the value of

1 (2 - ω) (2 - ω^{2}) + 2 (3 - ω) (3 - ω^{2}) + . . . . . + (n - 1) (n - ω) (n - ω^{2})

, is

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

n

is the smallest natural number such that

n + 2 n + 3 n + \dots . + 99 n

is a perfect square, then the number of digits in

n^{2}

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum

\frac{3 \times 1^{3}}{1^{2}} + \frac{5 \times (1^{3} + 2^{3})}{1^{2} + 2^{2}} + \frac{7 \times (1^{3} + 2^{3} + 3^{3})}{1^{2} + 2^{2} + 3^{2}} + . . . . .

upto

10^{t h}

term is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If, for a positive integer

n

, the quadratic equation,

x (x + 1) + (x + 1) (x + 2) + . .. + (x + \bar{n - 1}) (x + n) = 10 n

has two consecutive integral solutions, then

n

is equal to:

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

Let

a_{i} = i + \frac{1}{i}

for

i = 1, 2, \dots ., 20

. Put

p = \frac{1}{20} (a_{1} + a_{2} + \dots + a_{20})

and

q = \frac{1}{20} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + \dots + \frac{1}{a_{20}}) .

Then

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of odd integers from

1

2001

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the series

(1 + 2) + (1 + 2 + 2^{2}) + (1 + 2 + 2^{2} + 2^{3}) + \dots

upto

n

terms is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of first

9

terms of the series

\frac{1^{3}}{1} + \frac{1^{3} + 2^{3}}{1 + 3} + \frac{1^{3} + 2^{3} + 3^{3}}{1 + 3 + 5} + . . .

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the first

20

terms of the series

1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the series

1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \dots

upto

11^{t h}

term is:

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of all positive integers

n

for which

\frac{1^{3} + 2^{3} + \dots + {(2 n)}^{3}}{1^{2} + 2^{2} + \dots + n^{2}}

is also an integer is

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the

24

terms of the series

\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + \dots .

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The values of

\sum_{n = 0}^{1947} (\frac{1}{2^{n} + \sqrt{2^{1947}}})

is equal to

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

Let

A_{1}, A_{2} \dots ., A_{m}

be non-empty subsets of

{1, 2, 3, \dots, 100}

satisfying the following conditions.

(1)

The numbers

|A_{1}|, |A_{2}|, \dots, |A_{m}|

are distinct ;

(2)

A_{1}, A_{2}, . ., A_{m}

are pairwise disjoint.

(Here

|A|

denotes the number of elements in the set

A

). Then the maximum possible value of

m

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The value of

\cot (Σ_{n = 1}^{23} \cot^{- 1} (1 + Σ_{k = 1}^{n} 2 k))

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

Let

S_{k} = \frac{1 + 2 + 3 + \dots + k}{k}

. If

S_{1}^{2} + S_{2}^{2} + \dots + S_{10}^{2} = \frac{5}{12} A

, then

A

is equal to :

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If

99

more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly

2

balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The value of

\sum_{r = 16}^{30} (r + 2) (r - 3)

is equal to:

Find the sum of the following series: 22+42+62+.........+202=?

Important Questions on Sequence and Series

Learn and Prepare for any exam you want !

Find the sum of the following series:

$(2^{2} + 4^{2} + 6^{2} + . . . . . . . . . + 20^{2}) = ?$