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Sum of Infinite Terms of a GP

If a > 0, b...

Question

If $a > 0, b > 0$ and $a + b = 4$ , find the maximum value of ab.

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Solution

given $a + b = 4$ , $a > 0, b > 0$
We know that for a finite number of terms (real)
A.M $\geq$ G.M
Consider the terms to be $a, b$
A.M $= \frac{a + b}{2}$ ,
G.M $= \sqrt{a b}$
A.M $\geq$ G.M
$\frac{a + b}{2} \geq \sqrt{a b}$
$2 \geq \sqrt{a b}$
$a b \leq 4$
Maximum value of $a b$ is $4$

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