GM-HM inequality states that for positive reals \(a_i\) then \(\sqrt[n]{a_1a_2 \ldots a_n}\geq \frac{n}{\frac {1}{a_1}+\frac{1}{a_2} \ldots \frac{1}{a_n}}\) where equality occurs when all \(a_i\) are equal. So the value is always greater or equal than \(\frac{7}{\frac {1-a_1}{a_1}+\frac{1-a_2}{a_2} \ldots \frac{1-a_n}{a_n}}\) to the power of seven and its minimum is when this equality occurs. Here I assume them as positive reals less than 1, because if not, then let some \(a_k>1\), then the value is negative and grows smaller to negative infinity as the choice of \(a_k\) grow bigger.

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestIt is minimum when a1=a2=a3=a4=a5=a6=a7=6/7 Which gives (6/7)^7 / { 1-6/7}^7 = (6/7)^7 / (1/7)^7 =6^7 So, ab=6*7=42

Log in to reply

GM-HM inequality states that for positive reals \(a_i\) then \(\sqrt[n]{a_1a_2 \ldots a_n}\geq \frac{n}{\frac {1}{a_1}+\frac{1}{a_2} \ldots \frac{1}{a_n}}\) where equality occurs when all \(a_i\) are equal. So the value is always greater or equal than \(\frac{7}{\frac {1-a_1}{a_1}+\frac{1-a_2}{a_2} \ldots \frac{1-a_n}{a_n}}\) to the power of seven and its minimum is when this equality occurs. Here I assume them as positive reals less than 1, because if not, then let some \(a_k>1\), then the value is negative and grows smaller to negative infinity as the choice of \(a_k\) grow bigger.

Log in to reply

nicely done Yong See F.

Log in to reply

I assumed 0 < ai < 1 because if any one ai=0 and any one ai>=1 then we can get - infinity

Log in to reply