\(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+103\times105}}}}\)
= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+(104-1)\times(104+1)}}}}\)
= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+104^2-1}}}}\)
= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\times104}}}\)
= \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103-1)(103+1)}}}\)
= \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103^2-1)}}}\)
= \(\sqrt{1+100\sqrt{1+101\times103}}\)
= \(\sqrt{1+100\sqrt{1+(102-1)(102+1)}}\)
= \(\sqrt{1+100\sqrt{1+102^2-1}}\)
= \(\sqrt{1+100\times102}\)
= \(\sqrt{1+(101-1)(101+1)}\)
= \(\sqrt{1+101^2-1}\)
= 101
