\( \sqrt{1+100 \sqrt{1+101 \sqrt{1+102 \sqrt{1+103 \times 105}}}} \)

1.	\( \sqrt{1+100 \sqrt{1+101 \sqrt{1+102 \sqrt{1+103 \times 105}}}} \)
Answer» \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+103\times105}}}}\) = \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+(104-1)\times(104+1)}}}}\) = \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+104^2-1}}}}\) = \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\times104}}}\) = \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103-1)(103+1)}}}\) = \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103^2-1)}}}\) = \(\sqrt{1+100\sqrt{1+101\times103}}\) = \(\sqrt{1+100\sqrt{1+(102-1)(102+1)}}\) = \(\sqrt{1+100\sqrt{1+102^2-1}}\) = \(\sqrt{1+100\times102}\) = \(\sqrt{1+(101-1)(101+1)}\) = \(\sqrt{1+101^2-1}\) = 101

Answer»

\(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+103\times105}}}}\)

= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+(104-1)\times(104+1)}}}}\)

= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\sqrt{1+104^2-1}}}}\)

= \(\sqrt{1+100\sqrt{1+101\sqrt{1+102\times104}}}\)

= \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103-1)(103+1)}}}\)

= \(\sqrt{1+100\sqrt{1+101\sqrt{1+(103^2-1)}}}\)

= \(\sqrt{1+100\sqrt{1+101\times103}}\)

= \(\sqrt{1+100\sqrt{1+(102-1)(102+1)}}\)

= \(\sqrt{1+100\sqrt{1+102^2-1}}\)

= \(\sqrt{1+100\times102}\)

= \(\sqrt{1+(101-1)(101+1)}\)

= \(\sqrt{1+101^2-1}\)

= 101

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