Question Number 101252 by Dwaipayan Shikari last updated on 01/Jul/20
$$\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}\sqrt{\mathrm{1}+\mathrm{6}\sqrt{\mathrm{1}+\mathrm{7}\sqrt{\mathrm{1}+\mathrm{8}..}}}}}}\infty=? \\ $$
Commented by Dwaipayan Shikari last updated on 01/Jul/20
$$\mathrm{5}=\sqrt{\mathrm{1}+\mathrm{24}} \\ $$$$\:\:=\sqrt{\mathrm{1}+\mathrm{4}.\mathrm{6}}=\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{35}}}=\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}.\mathrm{7}}}=\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}\sqrt{\mathrm{1}+\mathrm{6}..}}} \\ $$$${So}\:\:\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}\sqrt{\mathrm{1}+\mathrm{6}\sqrt{\mathrm{1}+\mathrm{7}..}}}}}=\sqrt{\mathrm{1}+\mathrm{2}.\mathrm{5}}=\sqrt{\mathrm{11}} \\ $$