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Question Number 212320 by RojaTaniya last updated on 10/Oct/24

a+b+c+d=2, a^2 +b^2 +c^2 +d^2 =2 a^3 +b^3 +c^3 +d^3 =−4, a^4 +b^4 +c^4 +d^4 =−6 find real value of a^(2023) +b^(2023) +c^(2023) +d^(2023) .

$$\:{a}+{b}+{c}+{d}=\mathrm{2},\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} =\mathrm{2} \\ $$$$\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} =−\mathrm{4},\:{a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} +{d}^{\mathrm{4}} =−\mathrm{6} \\ $$$$\:{find}\:{real}\:{value}\:{of}\:{a}^{\mathrm{2023}} +{b}^{\mathrm{2023}} +{c}^{\mathrm{2023}} +{d}^{\mathrm{2023}} . \\ $$

Commented by mr W last updated on 10/Oct/24

=2^(1012)

$$=\mathrm{2}^{\mathrm{1012}} \\ $$

Answered by mr W last updated on 10/Oct/24

p_1 =e_1 =2 p_2 =e_1 p_1 −2e_2 =2 ⇒e_2 =1 p_3 =e_1 p_2 −e_2 p_1 +3e_3 =−4 ⇒e_3 =−2 p_4 =e_1 p_3 −e_2 p_2 +e_3 p_1 −4e_4 =−6 ⇒e_4 =−2 p_n =r_1 ^n +r_2 ^n +r_3 ^n +r_4 ^n r_(1−4) are roots of x^4 −2x^3 +x^2 +2x−2=0 (x^2 −1)(x^2 −2x+2)=0 r_(1,2) =±1, r_(3,4) =1±i=(√2) (cos (π/4)±i sin (π/4)) ⇒p_n =1+(−1)^n +2^((n/2)+1) cos ((nπ)/4) ⇒p_(2023) =1+(−1)^(2023) +2^(((2023)/2)+1) cos ((2023π)/4) =1−1+2^(1012) (√2) cos (π/4)=2^(1012) ✓

$${p}_{\mathrm{1}} ={e}_{\mathrm{1}} =\mathrm{2} \\ $$$${p}_{\mathrm{2}} ={e}_{\mathrm{1}} {p}_{\mathrm{1}} −\mathrm{2}{e}_{\mathrm{2}} =\mathrm{2}\:\Rightarrow{e}_{\mathrm{2}} =\mathrm{1} \\ $$$${p}_{\mathrm{3}} ={e}_{\mathrm{1}} {p}_{\mathrm{2}} −{e}_{\mathrm{2}} {p}_{\mathrm{1}} +\mathrm{3}{e}_{\mathrm{3}} =−\mathrm{4}\:\Rightarrow{e}_{\mathrm{3}} =−\mathrm{2} \\ $$$${p}_{\mathrm{4}} ={e}_{\mathrm{1}} {p}_{\mathrm{3}} −{e}_{\mathrm{2}} {p}_{\mathrm{2}} +{e}_{\mathrm{3}} {p}_{\mathrm{1}} −\mathrm{4}{e}_{\mathrm{4}} =−\mathrm{6}\:\Rightarrow{e}_{\mathrm{4}} =−\mathrm{2} \\ $$$${p}_{{n}} ={r}_{\mathrm{1}} ^{{n}} +{r}_{\mathrm{2}} ^{{n}} +{r}_{\mathrm{3}} ^{{n}} +{r}_{\mathrm{4}} ^{{n}} \\ $$$${r}_{\mathrm{1}−\mathrm{4}} \:{are}\:{roots}\:{of} \\ $$$${x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{2}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right)=\mathrm{0} \\ $$$${r}_{\mathrm{1},\mathrm{2}} =\pm\mathrm{1},\:{r}_{\mathrm{3},\mathrm{4}} =\mathrm{1}\pm{i}=\sqrt{\mathrm{2}}\:\left(\mathrm{cos}\:\frac{\pi}{\mathrm{4}}\pm{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{4}}\right) \\ $$$$\Rightarrow{p}_{{n}} =\mathrm{1}+\left(−\mathrm{1}\right)^{{n}} +\mathrm{2}^{\frac{{n}}{\mathrm{2}}+\mathrm{1}} \mathrm{cos}\:\frac{{n}\pi}{\mathrm{4}} \\ $$$$\Rightarrow{p}_{\mathrm{2023}} =\mathrm{1}+\left(−\mathrm{1}\right)^{\mathrm{2023}} +\mathrm{2}^{\frac{\mathrm{2023}}{\mathrm{2}}+\mathrm{1}} \mathrm{cos}\:\frac{\mathrm{2023}\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:=\mathrm{1}−\mathrm{1}+\mathrm{2}^{\mathrm{1012}} \sqrt{\mathrm{2}}\:\mathrm{cos}\:\frac{\pi}{\mathrm{4}}=\mathrm{2}^{\mathrm{1012}} \:\:\checkmark \\ $$

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