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Question Number 161102 by cortano last updated on 12/Dec/21

Commented by Abdulazizov last updated on 12/Dec/21


	Answered by mr W last updated on 13/Dec/21

	$p_{n} = a^{n} + b^{n} + c^{n}$ $p_{1} = e_{1} = 4$ $p_{2} = e_{1} p_{1} - 2 e_{2} \Rightarrow 10 = 4 \times 4 - 2 e_{2} \Rightarrow e_{2} = 3$ $p_{3} = e_{1} p_{2} - e_{2} p_{1} + 3 e_{3} \Rightarrow 22 = 4 \times 10 - 3 \times 4 + 3 e_{3} \Rightarrow e_{3} = - 2$ $p_{4} = e_{1} p_{3} - e_{2} p_{2} + e_{3} p_{1} = 4 \times 22 - 3 \times 10 - 2 \times 4 = 50$ $. . .$ $a, b, c a r e r o o t s o f$ $r^{3} - 4 r^{2} + 3 r + 2 = 0$ $(r - 2) (r^{2} - 2 r - 1) = 0$ $r_{1} = 2, r_{2, 3} = 1 \pm \sqrt{2}$ $i . e . (a, b, c) = (2, 1 + \sqrt{2}, 1 - \sqrt{2})$ $\Rightarrow p_{n} = 2^{n} + {(1 + \sqrt{2})}^{n} + {(1 - \sqrt{2})}^{n}$
	Commented by Rasheed.Sindhi last updated on 14/Dec/21

	$S i r, h o w - 2 c o m e i n^{'} p_{2} = e_{1} p_{1} - 2 e_{2}^{'}$ $(3 r d l i n e) ?$
	Commented by mr W last updated on 14/Dec/21

	$g e n e r a l l y$ $p_{n} = e_{1} p_{n - 1} - e_{2} p_{n - 2} + e_{3} p_{n - 3} - . . . + {(- 1)}^{n} e_{n - 1} p_{1} + {(- 1)}^{n + 1} {n e}_{n}$ $s e e {N e w t o n}^{'} s i d e n t i t i e s .$ $c e r t a i n l y w e c a n a l s o c h e c k i t :$ ${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$ $e_{1} p_{1} = p_{2} + 2 e_{2}$ $\Rightarrow p_{2} = e_{1} p_{1} - 2 e_{2}$
	Commented by Rasheed.Sindhi last updated on 14/Dec/21

	$T han k s a lOt S ir!$
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