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

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The number of isosceles triangle with integer sides, if no side exceeds $2022$ is

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Important Questions on Sequences and Series

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Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

The values of

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

is equal to

EASY

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

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 :

HARD

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

The sum of odd integers from

1

2001

MEDIUM

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

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:

EASY

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

The sum of the

24

terms of the series

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

HARD

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

The value of

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

HARD

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

\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>Sums of Squares and Cubes of the First n Natural Numbers

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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 :

MEDIUM

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

The largest perfect square that divides

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

is-

HARD

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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>Sums of Squares and Cubes of the First n Natural Numbers

The value of

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

is equal to:

HARD

Mathematics>Algebra>Sequences and Series>Sums of Squares and Cubes of the First n Natural Numbers

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} + . . .

The number of isosceles triangle with integer sides, if no side exceeds 2022 is

Important Questions on Sequences and Series

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The number of isosceles triangle with integer sides, if no side exceeds $2022$ is