Sums of Squares and Cubes of the First n Natural Numbers

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

Find the sum, $6+8+10+\cdots +72$.

Find the sum, $20+22+24+\cdots +72$.

Find the sum, $20+22+24+\cdots +96$.

Find the sum, $18+20+22+24+\cdots +102$.

Find the sum, $18+20+22+24+\cdots +60$.

Find the sums up to the $\mathrm{nth}$ term of following series 

$\frac{1^3}{2}+\frac{1^3+2^3}{2+4}+\frac{1^3+2^3+3^3}{2+4+6}+...$

Find the sums up to the $\mathrm{nth}$ term of following series 

$3.1^2+4.2^2+5.3^2+...$

Find the sums up to the $\mathrm{nth}$ term of following series 

$3^2+5^2+7^2+...$

Consider two sequence such that $a_1=2$ and $a_{r+1}=2a_r\forall r\in N$ and such that $\sum _{r=1}^n{b}_r=\frac{n(n+1)(2n+1)}{6}\forall n\in N,$then number of possible value(s) of $n$ for which $a_n=b_n,$ is-

The sum of $1^3+2^3+3^3+\dots \dots +{25}^3=$
 

The sum of cubes of first $n$ natural numbers is _____ of sum of the first $n$ natural numbers.

If the sum of first n positive integers is $\frac{1}{5}$ times the sum of their squares, then $\frac{n}{2}$ equals :

$\sum _{\mathrm{r}=1}^{\mathrm{n}}\mathrm{r}(\mathrm{r}+1)+\sum _{\mathrm{r}=1}^{\mathrm{n}}(\mathrm{r}+1)(\mathrm{r}+2)=\frac{\left(\mathrm{n}+{\mathrm{a}}_1\right)\left(\mathrm{n}+{\mathrm{a}}_2\right)\left(2\mathrm{n}+{\mathrm{a}}_3\right)}{3}-2$ :

Which of the following is compound?

$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots n$ terms $=$

The sum of the series $3^2+6^2+9^2+{12}^2+$ ... up to $n$ terms is $S_n,$ then $\frac{1}{41}{S}_{20}$ is equal to

Find the sum of the series $1^2+2^2-3^2+4^2+5^2-6^2+7^2+8^2-9^2+\dots +$ upto $\left(30\right)$ terms

If $\left(1+3+5+\dots +p\right)+\left(1+3+5+\dots +q\right)=\left(1+3+5+\dots +r\right)$ , where each set of parentheses contains the sum of consecutive odd integers as shown, the smallest possible value of $p+q+r$, ( where $p>6$ ) is

Find the value of the series $\mathrm{ }{1}^3-2^3+3^3-4^3\dots .$ is