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If $$ \frac{b_1}{a_1}, \frac{b_2}{a_2}, \dots, \frac{b_n}{a_n} $$ lies within $[\alpha, \beta]$, does $\frac{\sum_{i=1}^nb_i}{\sum_{i=1}^na_i}$ lies within the same range? (Given that $\sum_{i=1}^na_i \neq 0$.) If the answer is no, what about the situation that $a_n\gt 0$?

Arthur
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