Đáp án:
$2)a)\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{15}}}}<4\\ b)\sqrt{2}+\sqrt{5}+\sqrt{15}+\sqrt{24}<13$
Giải thích các bước giải:
$1)\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2018}}}}\\ \sqrt{2017\sqrt{2018}}<\sqrt{2017.2019}=\sqrt{(2018-1)(2018+1)}=\sqrt{2018^2-1}<2018\\ \sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2018}}}}\\ =\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2016\sqrt{2017\sqrt{2018}}}}}}\\ <\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2017.2019}}}}\\ <\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2016.2018}}}}\\ <\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{2015.2017}}}}\\\cdots \\< \sqrt{2.4}\\<3\\ 2)\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{15}}}}\\ <\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{16}}}}\\ =\sqrt{12+\sqrt{12+\sqrt{12+4}}}\\ =\sqrt{12+\sqrt{12+4}}\\ =\sqrt{12+4}\\ =4\\ \Rightarrow \sqrt{12+\sqrt{12+\sqrt{12+\sqrt{15}}}}<4\\ b)\sqrt{15}<\sqrt{16}=4\\ \sqrt{24}<\sqrt{25}=5\\ (\sqrt{2}+\sqrt{5})^2=7+2\sqrt{10}<7+2\sqrt{16}=15<16\\ \Rightarrow \sqrt{2}+\sqrt{5}<\sqrt{16}=4\\ \Rightarrow \sqrt{2}+\sqrt{5}+\sqrt{15}+\sqrt{24}<4+4+5=13$
