$$$\sum_{j=L}^{R} \sum_{k=j+1}^{R}(A_j*A_k)=((\sum_{j=L}^{R} A_j)^2-\sum_{j=L}^{R}A_j^2)/2$$$, so you can maintain two segment trees, one for $$$(\sum_{j=L}^{R} A_j)^2$$$, and the second for $$$\sum_{j=L}^{R}A_j^2$$$. To update first segment tree, you will need to maintain $$$\sum_{j=L}^{R} A_j$$$.

Represent the summation in a simpler way and then it should become trivial.