Let a 1 , a 2 , … … a 100 be non-zero real numbers such that a 1 + a 2 + … + a 100 = 0 , Then

∑ i = 1 100 a i 2 a i > 0 and ∑ i = 1 100 a i 2 - a i < 0

∑ i = 1 100 a i 2 a i ≥ 0 and ∑ i = 1 100 a i 2 - a i ≥ 0

∑ i = 1 100 a i 2 a i ≤ 0 and ∑ i = 1 100 a i 2 - a i ≤ 0

The sign of ∑ i = 1 100 a i 2 a i or ∑ i = 1 100 a i 2 - a i depends on the choice of a i ′ s

EASY

Earn 100

Tile sum of the first five terms of an A.P. is one-sixth the sum of next five terms. If the first term of the A.P. is $5$ , find the sum of first $30$ terms of the A.P.

Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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:

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The value of

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

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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} + . . .

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The sum of odd integers from

1

2001

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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}

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Let

S_{n} = \frac{1}{1^{3}} + \frac{1 + 2}{1^{3} + 2^{3}} + \frac{1 + 2 + 3}{1^{3} + 2^{3} + 3^{3}} + \dots + \frac{1 + 2 + \dots, + n}{1^{3} + 2^{3} + \dots n^{3}}

. If

100 S_{n} = n,

then

n

is equal to:

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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>Sequences and Series>Introduction to Sequences and 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

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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>Sequences and Series>Introduction to Sequences and Series

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

then

k

is equal to:

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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>Sequences and Series>Introduction to Sequences and Series

The values of

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

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Let

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

be non-zero real numbers such that

a_{1} + a_{2} + \dots + a_{100} = 0,

Then

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Let

a, b, c, d

be real numbers such that

\sum_{k = 1}^{n} (a k^{3} + b k^{2} + c k + d) = n^{4}

For every natural number

n

. Then

|a| + |b| + |c| + |d|

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The largest perfect square that divides

2014^{3} - 2013^{3} + 2013^{3} - 2011^{3} + \dots + 2^{3} - 1^{3}

is-

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The sum of the

24

terms of the series

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The value of

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

is equal to:

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and 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 :

Tile sum of the first five terms of an A.P. is one-sixth the sum of next five terms. If the first term of the A.P. is 5, find the sum of first 30 terms of the A.P.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Tile sum of the first five terms of an A.P. is one-sixth the sum of next five terms. If the first term of the A.P. is $5$ , find the sum of first $30$ terms of the A.P.