The sum of first 45 natural numbers is :

Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of all positive integers $n$ for which $\frac{1^3+2^3+\dots +{\left(2n\right)}^3}{1^2+2^2+\dots +n^2}$ is also an integer is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If, for a positive integer $n$, the quadratic equation,

$x\left(x+1\right)+\left(x+1\right)\left(x+2\right)+...+\left(x+\overline{n-1}\right)\left(x+n\right)=10n$ 

has two consecutive integral solutions, then $n$ is equal to:

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the first $20$ terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\dots$ is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The value of  $\cot \left(\underset{n=1}{\overset{23}{\Sigma }}{\cot }^{-1}\left(1+\underset{k=1}{\overset{n}{\Sigma }}2k\right)\right)$  is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of first $9$ terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+...$ is

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of odd integers from $1$ to $2001$ is

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

if $n$ is the smallest natural number such that $n+2n+3n+\dots .+99n$ is a perfect square, then the number of digits in $n^2$ is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous 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>Sequence and Series>Miscellaneous Series

Let $a_i=i+\frac{1}{i}$ for $i=1,2, \dots .,20$ . Put $p=\frac{1}{20}\left(a_1+a_2+\dots +a_{20}\right)$ and $q=\frac{1}{20}\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots +\frac{1}{a_{20}}\right).$ Then

EASY

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If $\omega$, is an imaginary cube root of$1$, then the value of $1\left(2-\omega \right)\left(2-{\omega }^2\right)+2\left(3-\omega \right)\left(3-{\omega }^2\right)+.....+\left(n-1\right)\left(n-\omega \right)\left(n-{\omega }^2\right)$, is

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If the sum of the first $15$ terms of the series ${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}_{ }^3+\dots$ is equal to $225\mathrm{K}$, then $K$ is equal to :

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

If ${\displaystyle \sum _{n=1}^5}\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{k}{3},$ then $k$ 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.

$\left(1\right)$ The numbers $\left|A_1\right|,\left|A_2\right|,\dots ,\left|A_m\right|$ are distinct ;

$\left(2\right)$ $A_1,A_2,..,A_m$ are pairwise disjoint.

(Here $\left|A\right|$ denotes the number of elements in the set $A$). Then the maximum possible value of $m$ is

HARD

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The values of $\sum _{n=0}^{1947}\left(\frac{1}{2^n+\sqrt{2^{1947}}}\right)$ is equal to

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous 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>Sequence and Series>Miscellaneous Series

Let $a, b, c, d$ be real numbers such that

$\sum _{k=1}^n\left(a{k}^3+b{k}^2+ck+d\right)=n^4$

For every natural number $n$. Then $\left|a\right|+\left|b\right|+\left|c\right|+\left|d\right|$ is equal to

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The largest perfect square that divides ${2014}^3-{2013}^3+{2013}^3-{2011}^3+\dots +2^3-1^3$ is-

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The sum of the $24$ terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\dots .$ Is

MEDIUM

Mathematics>Algebra>Sequence and Series>Miscellaneous Series

The value of $\sum _{r=16}^{30}\left(r+2\right)(r-3)$ is equal to:

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 :