<span>X+Y=a.XY=bのとき、次の式をa,bで表せ。</span><br><span>X^4+Y^4</span><br><span>＊＊＊＊＊＊＊＊</span><span><br></span><span>X,Yを解にもつ２次方程式</span><span>λ^2-aλ＋ｂ＝０</span><span>λ^2=aλ-b</span><span>λ^nをかければ</span><span>λ^(n+2)=aλ^(n+1)-bλ^n</span><span>にX,Yを代入して足せば</span><span>n=0:X^2+Y^2=a(X+Y)-2b=a^2-2b</span><span>n=1 X^3+Y^3=a(X^2+Y^2)-b(X+Y)=a(a^2-2b)-ba=a^3-3ab</span><span>n=2:X^4+Y^4=a(X^3+Y^3)-b(X^2+Y^2)=a(a^3-3ab)-b(a^2-2b)=a^4-4a^2b+2b^2</span><span>n=3:X^5+Y^5=a(X^4+Y^4)-b(X^3+Y^3)=a(a^4-4a^2b+2b^2)-b(a^3-3ab)=a^5-5a^3b+5ab^2</span><span>....</span><span><br></span>
