Safflower oil fractions

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approximating square root ask question asked today active today viewed times $egingroup$ we define recursive sequences $a_{n+ }= + rac { +a_n}$, $a_ = $ and $b_{n+ }= rac{b_n^ + }{ b_n}$, $b_ = $. i wish to show that $b_{n+ }=a_{ ^n}$. this can be proven using closed forms expressions related to continued fractions

be expressed as $$a_n=\sqrt \cdot rac{( +\sqrt )^n +( -\sqrt )^n}{( +\sqrt )^n - ( -\sqrt )^n}$$ on the other hand, we can prove inductively that $$ rac{b_{n+ }-\sqrt }{b_{n+ }+\sqrt }=\left( rac{ -\sqrt }{ +\sqrt } ight)^{ ^n}$$ so the relation $a_{ ^n}=b_{n+ }$ can be deduced by expanding the fractions...

https://math.stackexchange.com/questions/3491183/relation-between-two-rational-sequences-approximating-square-root-2

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