Let a1,a2,...,an be real numbers such that a21+2a22+⋯+na2n=1.
Find the maximum value of (a1+2a2+⋯+nan)2, in terms of n
I originally got the answer of 1 but it was wrong! Thanks for helping!
See my previous answer to your a12 question. It should be a similar (if not the exact same) process with Cauchy Schwarz.
Edit: my previous answer got large ai language modeled.
Here is how to start with cauchy: xi=√i⟹∑20i=1xi=√1+√2+...+√20⟹∑20i=1x2i=1+2+...+20. Cauchy says sum of x^2 * sum of y^2 >= sum of (xy)^2.