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Q.
If $x_1>0, i=1,2, \ldots, 50$ and $x_1+x_2+\ldots+x_{50}=50$, then the minimum value of $\frac{1}{x_1}+\frac{1}{x_2}+\ldots . .+\frac{1}{x_{50}}$ equals
Sequences and Series
Solution:
$x _1+ x _2+ x _3+\ldots \ldots \ldots \ldots+ x _{50}=50$
$AM \geq HM$
$\frac{ x _1+ x _2+\ldots \ldots . .+ x _{50}}{50} \geq \frac{1}{\frac{\frac{1}{ x _1}+\frac{1}{ x _2}+\frac{1}{ x _3}+\ldots .+\frac{1}{ x _{50}}}{50}}$
$\Rightarrow \frac{x_1+x_2+\ldots \ldots \ldots+x_{50}}{50} \geq \frac{50}{\frac{1}{x_1}+\frac{1}{x_2}+\ldots \ldots . .+\frac{1}{x_{50}}} $
$\Rightarrow \frac{1}{x_1}+\frac{1}{x_2}+\ldots \ldots . .+\frac{1}{x_{50}} \geq 50$
so minimum value of $\frac{1}{x_1}+\frac{1}{x_2}+\ldots \ldots . .+\frac{1}{x_{50}}=50$