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Question Number 157628 by mr W last updated on 25/Oct/21

find (C_0 ^(100) )^2 +(C_2 ^(100) )^2 +(C_4 ^(100) )^2 +(C_6 ^(100) )^2 +...+(C_(100) ^(100) )^2 =?

$${find} \\ $$$$\left({C}_{\mathrm{0}} ^{\mathrm{100}} \right)^{\mathrm{2}} +\left({C}_{\mathrm{2}} ^{\mathrm{100}} \right)^{\mathrm{2}} +\left({C}_{\mathrm{4}} ^{\mathrm{100}} \right)^{\mathrm{2}} +\left({C}_{\mathrm{6}} ^{\mathrm{100}} \right)^{\mathrm{2}} +...+\left({C}_{\mathrm{100}} ^{\mathrm{100}} \right)^{\mathrm{2}} =? \\ $$

Answered by mindispower last updated on 25/Oct/21

Σ_(k=0) ^(100) (C_(2k) ^(100) )^2 (1+ix)^(100) (1−ix)^(100) =Σ_(k=0) ^(100) C_(100) ^k (ix)^k Σ_(n=0) ^(100) C_n ^(100) (−ix)^n =(1+x^2 )^(100) =Σ_(k=0) ^(100) C_(100) ^k (ix)^k .Σ_(n=0) ^(100) C_n ^(100) (−ix)^n n+k=100 ⇒C_(100) ^(50) x^(100) =Σ_(k=0) ^(100) C_(100) ^k (ix)^k .C_(100−k) ^(100) (−i)^(100−k) x^(100−k) ⇒Σ_(k=0) ^(100) C_k ^(100) .C_(100−k) ^(100) (−1)^k x^(100) ⇒Σ_(k=0) ^(100) (−1)^k (C_k ^(100) )^2 =C_(50) ^(100) (1+x)^(100) (1+x)^(100) =Σ_(k=0) ^(100) C_k ^(100) x^k Σ_(n=0) ^(100) C_n ^(100) x^n C_(100) ^(200) x^(100) =Σ_(k=0) ^(100) (C_k ^(100) )^2 Σ(C_(2k) ^(100) )^2 =(1/2)Σ_(k=0) ^(100) ((C_k ^(100) )^2 +(−1)^k (C_k ^(100) )^2 ) Σ_(k=0) ^(50) (C_(2k) ^(100) )=(1/2)(C_(100) ^(200) +C_(50) ^(100) )

$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\left({C}_{\mathrm{2}{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} \\ $$$$\left(\mathrm{1}+{ix}\right)^{\mathrm{100}} \left(\mathrm{1}−{ix}\right)^{\mathrm{100}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{\mathrm{100}} ^{{k}} \left({ix}\right)^{{k}} \underset{{n}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{{n}} ^{\mathrm{100}} \left(−{ix}\right)^{{n}} \\ $$$$=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{100}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{\mathrm{100}} ^{{k}} \left({ix}\right)^{{k}} .\underset{{n}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{{n}} ^{\mathrm{100}} \left(−{ix}\right)^{{n}} \\ $$$${n}+{k}=\mathrm{100} \\ $$$$\Rightarrow{C}_{\mathrm{100}} ^{\mathrm{50}} {x}^{\mathrm{100}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{\mathrm{100}} ^{{k}} \left({ix}\right)^{{k}} .{C}_{\mathrm{100}−{k}} ^{\mathrm{100}} \left(−{i}\right)^{\mathrm{100}−{k}} {x}^{\mathrm{100}−\boldsymbol{{k}}} \\ $$$$\Rightarrow\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{{k}} ^{\mathrm{100}} .{C}_{\mathrm{100}−\boldsymbol{{k}}} ^{\mathrm{100}} \left(−\mathrm{1}\right)^{\boldsymbol{{k}}} \boldsymbol{{x}}^{\mathrm{100}} \\ $$$$\Rightarrow\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \left({C}_{{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} ={C}_{\mathrm{50}} ^{\mathrm{100}} \\ $$$$\left(\mathrm{1}+{x}\right)^{\mathrm{100}} \left(\mathrm{1}+{x}\right)^{\mathrm{100}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{{k}} ^{\mathrm{100}} {x}^{{k}} \underset{{n}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}{C}_{{n}} ^{\mathrm{100}} {x}^{{n}} \\ $$$${C}_{\mathrm{100}} ^{\mathrm{200}} {x}^{\mathrm{100}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\left({C}_{{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} \\ $$$$\Sigma\left({C}_{\mathrm{2}{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\left(\left({C}_{{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} +\left(−\mathrm{1}\right)^{{k}} \left({C}_{{k}} ^{\mathrm{100}} \right)^{\mathrm{2}} \right) \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{50}} {\sum}}\left({C}_{\mathrm{2}{k}} ^{\mathrm{100}} \right)=\frac{\mathrm{1}}{\mathrm{2}}\left({C}_{\mathrm{100}} ^{\mathrm{200}} +{C}_{\mathrm{50}} ^{\mathrm{100}} \right) \\ $$$$ \\ $$$$ \\ $$

Commented by mr W last updated on 26/Oct/21

great! thanks sir!

$${great}!\:{thanks}\:{sir}! \\ $$

Commented by mindispower last updated on 26/Oct/21

withe pleasur sir Have a nice day

$${withe}\:{pleasur}\:{sir} \\ $$$${Have}\:{a}\:{nice}\:{day} \\ $$

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