Question
Prove that $\left(a_1+a_2+\cdots+a_n\right)^2 \leq n\left(a_1^2+a_2^2+\cdots+a_n^2\right)$ for any real numbers $a_1, \ldots, a_n$. When does equality hold?
Step 1
\[ \left(a_1 + a_2 + \cdots + a_n\right)^2 = a_1^2 + a_2^2 + \cdots + a_n^2 + 2\sum_{i < j} a_i a_j \] Here, the sum \(2\sum_{i < j} a_i a_j\) represents the sum of all products of pairs of different \(a_i\)'s. Show more…
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