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Question Number 147543 by alcohol last updated on 21/Jul/21

$$\underset{m=1}{\prod ^n}{\left(\frac{1}{2}\right)}^m$$

Answered by Canebulok last updated on 21/Jul/21

$$$$$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:$$$$let:$$$$\Rightarrow \underset{m=1}{\prod ^n}{\left(\frac{1}{2}\right)}^m=\phi$$$$Bytakingthenaturallogofbothsides,$$$$\Rightarrow \underset{m=1}{\sum ^n}Ln\left(\frac{1}{2}\right)\cdot \left(m\right)=Ln\left(\phi \right)$$$$\Rightarrow Ln\left(\frac{1}{2}\right)\cdot \underset{m=1}{\sum ^n}\left(m\right)=Ln\left(\phi \right)$$$$\Rightarrow Ln\left(\frac{1}{2}\right)\cdot \left(\frac{n\cdot (n+1)}{2}\right)=Ln\left(\phi \right)$$$$\because$$$$\Rightarrow \phi ={\left(\frac{1}{2}\right)}^{\frac{n\cdot (n+1)}{2}}$$$$$$

Answered by gsk2684 last updated on 21/Jul/21

$$\underset{m=1}{\prod ^n}{\left(\frac{1}{2}\right)}^m$$$$={\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^3....{\left(\frac{1}{2}\right)}^n$$$$={\left(\frac{1}{2}\right)}^{1+2+3+...+n}$$$$={\left(\frac{1}{2}\right)}^{\frac{n(n+1)}{2}}$$

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