If $t_n=n(n!),$ then the value of $\underset{n=1}{\overset{15}{\mathrm{\Sigma }}}{t}_n$ is

Important Questions on Sequences and Series

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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:

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special Series

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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>Sequences and Series>Sum to n Terms of Special Series

The sum of the series $(1+2)+\left(1+2+2^2\right)+\left(1+2+2^2+2^3\right)+\dots$  upto $n$ terms is 

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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

EASY

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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

EASY

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special Series

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

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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 :

EASY

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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>Sequences and Series>Sum to n Terms of Special 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

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special Series

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

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special Series

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

HARD

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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>Sequences and Series>Sum to n Terms of Special 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>Sequences and Series>Sum to n Terms of Special 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:

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special 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 ${10}^{th}$ term is

MEDIUM

Mathematics>Algebra>Sequences and Series>Sum to n Terms of Special Series

The sum of the series $1+2\times 3+3\times 5+4\times 7+\dots$ upto ${11}^{th}$ term is: