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Question Number 141535 by mohammad17 last updated on 20/May/21

how can solve this with by steps please ((Γ((3/2))Γ((7/2)))/(2Γ(5))) i want solution step by step because i understand this?

$${how}\:{can}\:{solve}\:{this}\:{with}\:{by}\:{steps}\:{please} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\mathrm{5}\right)}\:{i}\:{want}\:{solution}\:{step}\:{by}\:{step} \\ $$$${because}\:{i}\:{understand}\:{this}? \\ $$$$ \\ $$

Commented by Dwaipayan Shikari last updated on 20/May/21

Γ((3/2))=Γ((1/2)+1)=(1/2)Γ((1/2))=((√π)/2) Γ((7/2))=Γ((5/2)+1)=(5/2)Γ((3/2)+1)=(5/2).(3/2)Γ((1/2)+1)=((5.3.1)/2^3 )Γ((1/2))=((15(√π))/8) ((Γ((3/2))Γ((7/2)))/(2Γ(5)))=((15π)/(32.4!))=((3.5π)/(3.256))=((5π)/(256))

$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$$$\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)=\Gamma\left(\frac{\mathrm{5}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}}{\mathrm{2}}.\frac{\mathrm{3}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}.\mathrm{3}.\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{15}\sqrt{\pi}}{\mathrm{8}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\mathrm{5}\right)}=\frac{\mathrm{15}\pi}{\mathrm{32}.\mathrm{4}!}=\frac{\mathrm{3}.\mathrm{5}\pi}{\mathrm{3}.\mathrm{256}}=\frac{\mathrm{5}\pi}{\mathrm{256}} \\ $$

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