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Question Number 116595 by ZiYangLee last updated on 05/Oct/20

If (1−2x+3x^2 )^(10) =a_0 +a_1 x+a_2 x^2 +...+a_(20) x^(20)   find the value of a_1 +a_2 +...+a_(20)

$$\mathrm{If}\:\left(\mathrm{1}−\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +...+{a}_{\mathrm{20}} {x}^{\mathrm{20}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{\mathrm{20}} \\ $$

Answered by mr W last updated on 05/Oct/20

put x=0:  (1−0+0)^(10) =a_0 +a_1 0+a_2 0+...+a_(20) 0  ⇒a_0 =1  put x=1:  (1−2+3)^(10) =a_0 +a_1 1+a_2 1+...+a_(20) 1  ⇒2^(10) =a_0 +a_1 +a_2 +...+a_(20)     ⇒a_1 +a_2 +...+a_(20) =2^(10) −a_0 =2^(10) −1

$${put}\:{x}=\mathrm{0}: \\ $$$$\left(\mathrm{1}−\mathrm{0}+\mathrm{0}\right)^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} \mathrm{0}+{a}_{\mathrm{2}} \mathrm{0}+...+{a}_{\mathrm{20}} \mathrm{0} \\ $$$$\Rightarrow{a}_{\mathrm{0}} =\mathrm{1} \\ $$$${put}\:{x}=\mathrm{1}: \\ $$$$\left(\mathrm{1}−\mathrm{2}+\mathrm{3}\right)^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} \mathrm{1}+{a}_{\mathrm{2}} \mathrm{1}+...+{a}_{\mathrm{20}} \mathrm{1} \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{\mathrm{20}} \\ $$$$ \\ $$$$\Rightarrow{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{\mathrm{20}} =\mathrm{2}^{\mathrm{10}} −{a}_{\mathrm{0}} =\mathrm{2}^{\mathrm{10}} −\mathrm{1} \\ $$

Commented by ZiYangLee last updated on 05/Oct/20

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