1^2 +2^2 +3^2 +5^2 +8^2 +13^2 +21^2 =?


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Question Number 208217 by mr W last updated on 07/Jun/24

$1^{2} + 2^{2} + 3^{2} + 5^{2} + 8^{2} + 13^{2} + 21^{2} = ?$
	Answered by A5T last updated on 07/Jun/24

	$T h i s i s t r i v i a l, b u t c o u l d b e i n t e r e s t i n g$ $w i t h a c o n n e c t i o n w i t h F i b o n a c c i s e q u e n c e .$ $f_{n} = f_{n - 1} + f_{n - 2}; f_{1} = f_{2} = 1, f_{3} = 2$ $f_{1}^{2} + f_{2}^{2} + . . . + f_{n}^{2} \overset{?}{=} f_{n} f_{n + 1} . . . (i)$ $I n d u c t i o n; b a s e c a s e :$ $f_{1}^{2} = f_{1} \times f_{2} = 1; f_{1}^{2} + f_{2}^{2} = f_{2} \times f_{3} = 2$ $S o, s u p p o s e \exists n_{1} (n_{1} = 1) s u c h t h a t (i) i s t r u e; w e$ $s h o w t h a t i t m u s t b e t r u e f o r n_{1} + 1; h e n c e f o r a l l$ $n ⩾ n_{1}$ $f_{1}^{2} + f_{2}^{2} + . . . + f_{n_{1}}^{2} = f_{n_{1}} f_{n_{1} + 1}$ $\Rightarrow f_{1}^{2} + f_{2}^{2} + . . . f_{n_{1}}^{2} + f_{n_{1} + 1}^{2} = f_{n_{1}} f_{n_{1} + 1} + f_{n_{1} + 1}^{2}$ $= f_{n_{1} + 1} (f_{n_{1}} + f_{n_{1} + 1}) = f_{n_{1} + 1} f_{n_{1} + 2}$ $\Rightarrow n_{1} + 1 i s t r u e; s o (i) i s t r u e f o r a l l n ⩾ 1$ $\Rightarrow 1^{2} + 1^{2} + 2^{2} + 3^{2} + 5^{2} + 8^{2} + 13^{2} + 21^{2} = 21 \times 34$ $\Rightarrow ? = (21 \times 34) - 1 = 713$
	Commented by mr W last updated on 07/Jun/24
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	Answered by zamin2001 last updated on 07/Jun/24

	$1^{2} + 1^{2} + 2^{2} + 3^{2} + 5^{2} + 8^{2} + 13^{2} + 21^{2} = 21 \times 34$ $\Rightarrow 1^{2} + 2^{2} + 3^{2} + 5^{2} + 8^{2} + 13^{2} + 21^{2} = 21 \times 34 - 1 = 713$
	Commented by mr W last updated on 07/Jun/24
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	Answered by A5T last updated on 07/Jun/24

	$A n o t h e r p r o o f (c o n s t r u c t i v e) :$ $f_{n - 1} f_{n + 1} = (f_{n + 1} - f_{n}) (f_{n} + f_{n - 1})$ $f_{n - 1} f_{n + 1} = f_{n} f_{n + 1} + f_{n + 1} f_{n - 1} - f_{n}^{2} - f_{n} f_{n - 1}$ $\Rightarrow f_{n}^{2} = f_{n} f_{n + 1} - f_{n} f_{n - 1} (w e g e t a t e l e s c o p i n g s u m)$ $\Rightarrow f_{1}^{2} + f_{2}^{2} + . . . + f_{n}^{2} = f_{n} f_{n + 1}$
	Commented by mr W last updated on 07/Jun/24
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