I am having trouble with taking a certain summation and finding an explicit value for the summation. The summation is: $$ S = \sum_{w=3}^a \lambda_w \sum_{m=w}^a \lambda_m $$ The only information available is \begin{align*} k = \sum_{v=1}^a \lambda_v \end{align*} Where $k$ is a positive integer. I am looking to find out if this has an generalized explicit value of this summation. I would also like to learn how i can do these types of problems in the future. Suggested tags are welcome.
WHAT I HAVE TRIED:
I know that $$ \sum_{m=w}^a \lambda_m = k-\sum_{m=1}^{w-1} \lambda_m $$ Yet once this is played out i end up with: \begin{align*} S = \sum_{w=3}^a \lambda_w (k-\sum_{m=1}^{w-1}\lambda_m) \qquad\qquad\qquad\qquad\quad \\ = (k-\lambda_1 -\lambda_2 - \lambda_3)(\lambda_1+\lambda_2+\lambda_3) - \sum_{w=4}^a \lambda_w\sum_{m=4}^{w-1} \lambda_m\qquad \end{align*} Which leaves me nowhere.
\begin{align*} S &= \sum_{w=3}^a \lambda_w (k-\sum_{m=1}^w\lambda_m) \qquad\qquad\qquad\qquad\quad \\ &= k^2-k(\lambda_1+\lambda_2)-\sum_{w=3}^a \sum_{m=1}^w\lambda_m\lambda_w\\ &=k^2-k(\lambda_1+\lambda_2)-\sum_{m=3}^a\sum_{w=m}^a \lambda_m\lambda_w -\lambda_1 k-\lambda_2k+(\lambda_1+\lambda_2)^2 \end{align*} You might recognize the double sum as $S$ itself. Hence
$$2S=k^2-2k(\lambda_1+\lambda_2)+(\lambda_1+\lambda_2)^2$$ $$2S=(k-(\lambda_1+\lambda_2))^2$$
P.S.: I don't know if the value $(\lambda_1+\lambda_2)$ is allowed to figure in the "generalized explicit value".