Miscellaneous Series

Find the sum of the following series:

$\left({11}^2+{12}^2+{13}^2+.........+{20}^2\right)=?$

Find the sum of first $9$ even natural numbers:

$2+4+6+8+10+12+14+16+18$

What is the sum of the series $\frac{1}{5\times 8}+\frac{1}{8\times 11}..............+\frac{1}{242\times 245}$?

If sum of first $n$ natural numbers is $\frac{1}{5}$ of the sum of their squares, then $n$ is _____

If the sum of first $n$ natural number is $\frac{1}{5}$ of the sum of their squares, then $n$ is ________.

Find the sum of the following series:

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

Find the sum of first $7$ odd natural numbers:

$1+3+5+7+9+11+13$

The sum of all the digits of the numbers from $1$ to $102$ is?

Find the sum of cubes of first $6$ natural numbers:

$21 22 23$ ................ $39$

$1 + 2 + 3 + 4 + .........+ 99 + 100 + 99 + .......+ 3 + 2 + 1$ is equal to ?

Find the sum of following series:

$\left(1^2+2^2+3^2+.........+{10}^2\right)=?$

$1+ 2 +3 ......... 32 +31 +30 ....... 3+ 2 + 1 =?$

Find the sum of squares of first $8$ natural numbers:

$1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2$

Mention any two properties of $\mathrm{Pi}$ Notation.

Using the fact that $\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$, the value of $\sum _{n=1}^{\infty }\frac{1}{{\left(2n+1\right)}^2}$ is -

The sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\mathrm{......}$ is

Let $S$ be the infinite sum given by $S=\sum _{n=0}^{\infty }\frac{a_n}{{10}^{2n}}$ where ${\left(a_n\right)}_n\ge 0$ is a sequence defined by $a_0=a_1=1$ and $a_j=20a_{j-1}-108a_{j-2}$ for $j\ge 2$. If $S$ is expressed in the form $\frac{a}{b}$, where $a,b$ are coprime positive integers, then $a$ equals

The mean of the squares of first $\text{'}n\text{'}$ natural numbers is

Given $\frac{\sin 1°}{\sin x°\sin \left(x+1\right)°}=\cot x°-\cot \left(x+1\right)°$, then the value of $\frac{1}{\sin 45°\sin 46°}+\frac{1}{\sin 46°\sin 47°}+\cdots +\frac{1}{\sin 89°\sin 90°}$ is$$