zmudz one year ago 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 $$(1a_1+2a_2+\cdots+na_n)^2?$$

$$\phi(x)=x^2$$ is concave, so by Jensen's inequality: $$\phi\left(\frac{\sum ia_i}{\sum i}\right)\le\frac{\sum i\phi(a_i)}{\sum i}\\\left(\frac{\sum ia_i}{n(n+1)/2}\right)^2\le\frac{\sum ia_i^2}{n(n+1)/2}$$ now we're told that $$\sum ia_i^2=1$$ so: $$\left(\sum ia_i\right)^2\le\frac{n(n+1)}2$$