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Question Number 30918 by naka3546 last updated on 28/Feb/18

Answered by ajfour last updated on 28/Feb/18

$$a+b+c=s_1=15$$$$a^2+b^2+c^2=s_2=107$$$$a^3+b^3+c^3=s_3=855$$$$a^5+b^5+c^5=s_5=?$$$$s_2{s}_3=s_5+a^2{b}^2(a+b)+b^2{c}^2(b+c)$$$$+c^2{a}^2(c+a)$$$$s_2{s}_3=s_5+s_1(a^2{b}^2+b^2{c}^2+c^2{a}^2)+$$$$-abc(ab+bc+ca)...\left(i\right)$$$$s_1^2=s_2+2(ab+bc+ca)$$$$\Rightarrow ab+bc+ca=\frac{s_1^2-s_2}{2}...\left(ii\right)$$$$\Rightarrow ab+bc+ca=\frac{225-107}{2}=59$$$$s_3=3abc+s_1(s_2-ab-bc-ca)$$$$abc=\frac{s_3-s_1(s_2-ab-bc-ca)}{3}$$$$\Rightarrow abc=\frac{855-15(107-59)}{3}$$$$=45$$$${(ab+bc+ca)}^2=a^2{b}^2+b^2{c}^2+c^2{a}^2$$$$+2abc(a+b+c)$$$${\left(59\right)}^2=a^2{b}^2+b^2{c}^2+c^2{a}^2+2\times 45\times 15$$$$\Rightarrow a^2{b}^2+b^2{c}^2+c^2{a}^2=2131$$$$\mathit{N}\mathit{o}\mathit{w}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\left(\mathit{i}\right),$$$${\mathit{s}}_2{\mathit{s}}_3={\mathit{s}}_5+{\mathit{s}}_1({\mathit{a}}^2{\mathit{b}}^2+{\mathit{b}}^2{\mathit{c}}^2+{\mathit{c}}^2{\mathit{a}}^2)$$$$-\mathit{a}\mathit{b}\mathit{c}(\mathit{a}\mathit{b}+\mathit{b}\mathit{c}+\mathit{c}\mathit{a})$$$$107\times 855={\mathit{s}}_5+15\times 2131-45\times 59$$$$\Rightarrow {\mathit{s}}_5=107\times 855+45\times 59-15\times 2131$$$${\mathit{s}}_5=62,175.$$

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