Question Number 157006 by amin96 last updated on 18/Oct/21
$$\sqrt{\mathrm{4}+\mathrm{27}\sqrt{\mathrm{4}+\mathrm{29}\sqrt{\mathrm{4}+\mathrm{31}\sqrt{\mathrm{4}+\ldots}}}}=? \\ $$$$ \\ $$
Answered by puissant last updated on 18/Oct/21
$${Sir}\:{use}\:{Ramanujan}\:{formula} \\ $$$${x}+\mathrm{4}=\sqrt{\mathrm{4}+{x}\sqrt{\mathrm{4}+\left({x}+\mathrm{2}\right)\sqrt{\mathrm{4}+\left({x}+\mathrm{4}\right)\sqrt{.....}}}} \\ $$$${for}\:{x}=\mathrm{27} \\ $$$$\Rightarrow\:\sqrt{\mathrm{4}+\mathrm{27}\sqrt{\mathrm{4}+\mathrm{29}\sqrt{\mathrm{4}+\mathrm{31}\sqrt{\mathrm{4}+...}}}}\:=\:\mathrm{31}.. \\ $$