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

Π_(m=1) ^n ((1/2))^m

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

Answered by Canebulok last updated on 21/Jul/21

   Solution:  let:  ⇒ Π_(m=1) ^n  ((1/2))^m  = ϕ  By taking the natural log of both sides,  ⇒ Σ_(m=1) ^n  Ln((1/2))∙(m) = Ln(ϕ)  ⇒ Ln((1/2))∙Σ_(m=1) ^n  (m) = Ln(ϕ)  ⇒ Ln((1/2))∙(((n∙(n+1))/2)) = Ln(ϕ)  ∵  ⇒ ϕ = ((1/2))^((n∙(n+1))/2)

$$\: \\ $$$$\boldsymbol{{Solution}}: \\ $$$${let}: \\ $$$$\Rightarrow\:\underset{{m}=\mathrm{1}} {\overset{{n}} {\prod}}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{m}} \:=\:\varphi \\ $$$${By}\:{taking}\:{the}\:{natural}\:{log}\:{of}\:{both}\:{sides}, \\ $$$$\Rightarrow\:\underset{{m}=\mathrm{1}} {\overset{{n}} {\sum}}\:{Ln}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\centerdot\left({m}\right)\:=\:{Ln}\left(\varphi\right) \\ $$$$\Rightarrow\:{Ln}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\centerdot\underset{{m}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left({m}\right)\:=\:{Ln}\left(\varphi\right) \\ $$$$\Rightarrow\:{Ln}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\centerdot\left(\frac{{n}\centerdot\left({n}+\mathrm{1}\right)}{\mathrm{2}}\right)\:=\:{Ln}\left(\varphi\right) \\ $$$$\because \\ $$$$\Rightarrow\:\varphi\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{{n}\centerdot\left({n}+\mathrm{1}\right)}{\mathrm{2}}} \\ $$$$\: \\ $$

Answered by gsk2684 last updated on 21/Jul/21

Π_(m=1) ^n ((1/2))^m   =((1/2))^1 ((1/2))^2 ((1/2))^3 ....((1/2))^n   =((1/2))^(1+2+3+...+n)   =((1/2))^((n(n+1))/2)

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

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