For any integer $n \geq 1$,$\sum_{K=1}^n K(K+2) =$

If $1^2 + 2^2 + 3^2 + \dots + 2009^2 = (2009)(335)(4019)$ and $(1)(2009) + 2(2008) + 3(2007) + \dots + 2009(1) = (2009)(335)(x)$,then $x$ is equal to:

$t_1, t_2, t_3, \ldots, t_{n}$ are positive integers,$S_{n} = t_1 + t_2 + t_3 + \ldots + t_{n}$. Given $S_1 = 1^2, S_2 = 3^2, S_3 = 6^2, S_4 = 10^2, S_5 = 15^2$. Following this pattern,if $S_{10} = k^2$,then $k =$

$\frac{10001 \times 100 !}{2 \times 1 !+5 \times 2 !+10 \times 3 !+\ldots+10001 \times 100 !}=$

If $1+\frac{\cos \theta}{2}+\frac{\cos 2 \theta}{4}+\frac{\cos 3 \theta}{8}+\ldots = \frac{a-2 \cos \theta}{5+b \cos \theta}$ for some $a, b \in R$,then $(a-b)^2=$

Find the $9^{\text{th}}$ term in the sequence whose $n^{\text{th}}$ term is given by $a_{n} = (-1)^{n-1} n^{3}$.