Solve for x: x^2 +x^2 (1−x^2 )+x^2 (1−x^2 )^2 +x^2 (1−x^2 )^3 +...+x^2 (1−x^2 )^(100) =1


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Question Number 208164 by Fridunatjan08 last updated on 06/Jun/24

$S o l v e f o r x :$ $x^{2} + x^{2} (1 - x^{2}) + x^{2} {(1 - x^{2})}^{2} + x^{2} {(1 - x^{2})}^{3} + . . . + x^{2} {(1 - x^{2})}^{100} = 1$
	Answered by A5T last updated on 06/Jun/24

	$x^{2} + x^{2} (1 - x^{2}) + x^{2} {(1 - x^{2})}^{2} + . . . + x^{2} {(1 - x^{2})}^{100} = 1$ $\Rightarrow x^{2} (1 - x^{2}) + x^{2} {(1 - x^{2})}^{2} + . . . + x^{2} {(1 - x^{2})}^{100} = 1 - x^{2}$ $D i v i d i n g t h r o u g h b y 1 - x^{2} [x \neq \underline{+} 1]$ $\Rightarrow x^{2} + x^{2} (1 - x^{2}) + x^{2} {(1 - x^{2})}^{2} + . . . + x^{2} {(1 - x^{2})}^{99} = 1$ $S u b t r a c t i n g x^{2} f r o m b o t h s i d e s, d i v i d i n g b y$ $(1 - x^{2}) a n d i t e r a t i n g$ $\Rightarrow x^{2} (1 - x^{2}) = 1 - x^{2} \Rightarrow x^{2} = 1 \Rightarrow x = \underline{+} 1$ $B u t b y a s s u m p t i o n t h a t x \neq \underline{+} 1 i n i t i a l l y$ $\Rightarrow x = \underline{+} 1$
	Commented by Fridunatjan08 last updated on 07/Jun/24

	$t h a n k s s i r .$
	Answered by Berbere last updated on 07/Jun/24
	$x^2 (Σ_(k=0) ^(100) (1−x^2 )^k )=1⇒x≠0 x^2 .((1−(1−x^2 )^(101) )/x^2 )=1 ⇔1−(1−x^2 )^(101) =1⇒x^2 =1⇒x∈{−1,1}$
	$x^{2} (\underset{k = 0}{\sum^{100}} {(1 - x^{2})}^{k}) = 1 \Rightarrow x \neq 0$ $x^{2} . \frac{1 - {(1 - x^{2})}^{101}}{x^{2}} = 1$ $\Leftrightarrow 1 - {(1 - x^{2})}^{101} = 1 \Rightarrow x^{2} = 1 \Rightarrow x \in {- 1, 1}$
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