Consider $p_1,p_2,...p_N$ are probabilities arranged in ascending order. $n_1, n_2,...n_N$ are numbers which are arranged in geometric progression.
I want to Maximize E= $\sum\limits_{i=1}^N p_i\cdot n_i$, subject to the constraint $\sum\limits_{i=1}^N n_i=L$
Here, $ n_1=a, n_2=ar, n_3=ar^2, ...,n_N=ar^{N-1} $
Hence, $a+ar+ ar^2+...+ar^{N-1}=L$ and
$E=p_1*a+p_2*ar+p_3*ar^2+..+p_N*ar^{N-1}$
How to find values of $a$ and $r$ which will maximize E.
Thanks, Kunal