+36 Let \(a_1,a_2,...,a_n\) be real numbers such that \(a_1^2+2a_2^2+\cdots+na_n^2 = 1.\)
Find the maximum value of \((a_1+2a_2+\cdots+na_n)^2,\) in terms of \(n\)
I originally got the answer of 1 but it was wrong! Thanks for helping!
+35 See my previous answer to your \(a_{12}\) 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: \(x_i = \sqrt{i} \implies \sum_{i=1} ^{20}{x_i} = \sqrt{1} + \sqrt{2} + ... + \sqrt{20} \implies \sum_{i=1} ^{20}{x_i^2} = 1 + 2 + ... + 20\). Cauchy says sum of x^2 * sum of y^2 >= sum of (xy)^2.
