Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 136098 by SOMEDAVONG last updated on 18/Mar/21

I).compute Σ_(k=0) ^(1000) C_(2000) ^(2k) .

$$\left.\mathrm{I}\right).\mathrm{compute}\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2000}} ^{\mathrm{2k}} . \\ $$

Answered by Olaf last updated on 18/Mar/21

(1−1)^(2000) = 0 = Σ_(p=0) ^(2000) (−1)^p C_p ^(2000) 0 = Σ_(k=0) ^(1000) (−1)^(2k) C_(2k) ^(2000) +Σ_(k=0) ^(999) (−1)^(2k+1) C_(2k+1) ^(2000) 0 = Σ_(k=0) ^(1000) C_(2k) ^(2000) −Σ_(k=0) ^(999) C_(2k+1) ^(2000) Σ_(k=0) ^(1000) C_(2k) ^(2000) = Σ_(k=0) ^(999) C_(2k+1) ^(2000) (1+1)^(2000) = 2^(2000) = Σ_(p=0) ^(2000) C_p ^(2000) 2^(2000) = Σ_(k=0) ^(1000) C_(2k) ^(2000) +Σ_(k=0) ^(999) C_(2k+1) ^(2000) 2^(2000) = Σ_(k=0) ^(1000) C_(2k) ^(2000) +Σ_(k=0) ^(1000) C_(2k) ^(2000) 2^(2000) = 2Σ_(k=0) ^(1000) C_(2k) ^(2000) Σ_(k=0) ^(1000) C_(2k) ^(2000) = (1/2)(2^(2000) ) = 2^(1999)

$$\left(\mathrm{1}−\mathrm{1}\right)^{\mathrm{2000}} \:=\:\mathrm{0}\:=\:\underset{{p}=\mathrm{0}} {\overset{\mathrm{2000}} {\sum}}\left(−\mathrm{1}\right)^{{p}} \mathrm{C}_{{p}} ^{\mathrm{2000}} \\ $$$$\mathrm{0}\:=\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{2}{k}} \mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} +\underset{{k}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{2}{k}+\mathrm{1}} \mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{2000}} \\ $$$$\mathrm{0}\:=\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} −\underset{{k}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{2000}} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} \:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{2000}} \\ $$$$ \\ $$$$\left(\mathrm{1}+\mathrm{1}\right)^{\mathrm{2000}} \:=\:\mathrm{2}^{\mathrm{2000}} \:=\:\:\underset{{p}=\mathrm{0}} {\overset{\mathrm{2000}} {\sum}}\mathrm{C}_{{p}} ^{\mathrm{2000}} \\ $$$$\mathrm{2}^{\mathrm{2000}} \:=\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} +\underset{{k}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{2000}} \\ $$$$\mathrm{2}^{\mathrm{2000}} \:=\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} +\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} \\ $$$$\mathrm{2}^{\mathrm{2000}} \:=\:\:\mathrm{2}\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{2000}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}^{\mathrm{2000}} \right)\:=\:\mathrm{2}^{\mathrm{1999}} \\ $$

Commented by SOMEDAVONG last updated on 18/Mar/21

THANKS TEACHER

$$ \\ $$$$\mathbb{THANKS}\:\:\:\mathbb{TEACHER} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com