Assume that $1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$ where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of...

Question

Assume that $1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$ where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of $(1a_1+2a_2+\cdots+na_n)^2?$

Thanks for all your help!

Answer 1

By the Cauchy Schwartz Inequality, $$\frac{n(n+1)}{2}=\left(\sum_{k=1}^n k a_k^2\right)\left(\sum_{k=1}^n\sum_{t=1}^k 1\right)\geq\left(\sum_{k=1}^n ka_k\right)^2$$ Hence, the maximum of $\left(\sum_{k-1}^nka_k\right)^2$ is $\frac{n(n+1)}{2}$.

Note: the summations above can also be written as $$\frac{n(n+1)}{2}=(a_1^2+a_2^2+a_2^2+a_3^2+a_3^2+a_3^2+...)(1+1+1+...)\geq(a_1+2a_2+....+na_n)^2$$