Simplify \(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}}.\)
\(S\ =\ \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{1}{\sqrt{n}+\sqrt{n+1}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{1}{\sqrt{n}+\sqrt{n+1}}\cdot \frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n}-\sqrt{n+1}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{\sqrt{n}-\sqrt{n+1}}{(n)-(n+1)}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{\sqrt{n}-\sqrt{n+1}}{-1}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\sqrt{n+1}-\sqrt{n}\\~\\ S\ =\ {\color{}\sqrt2}-\sqrt1\ {\color{}+\sqrt3-\sqrt2+\sqrt4-\sqrt3+\sqrt5-\sqrt4+\dots+\sqrt{99}-\sqrt{98}}+\sqrt{100}\ {\color{}-\ \sqrt{99}}\\~\\ S\ =\ {\color{gray}\sqrt2}-\sqrt1\ {\color{gray}+\sqrt3-\sqrt2+\sqrt4-\sqrt3+\sqrt5-\sqrt4+\dots+\sqrt{99}-\sqrt{98}}+\sqrt{100}\ {\color{gray}-\ \sqrt{99}}\\~\\ S\ =\ -\sqrt1+\sqrt{100}\\~\\ S\ =\ -1+10\\~\\ S\ =\ 9 \)_
.1 (1 - √2) (1 - √2)
______________ =___________ = ( √2 - 1)
(1 + √2) ( 1 - √2) -1
1 ( √2 - √3) (√2 - √3)
______________ =___________ = ( √3 - √2)
(√2+ √3) ( √2 - √3) -1
1 ( √3 - √4) (√3 - √4)
_______________ = ___________ = ( √4 - √3)
(√3 + √4) ( √3 - √4) -1
So we have
-1 + ( √2 - √2) + ( √3 - √3) + ....... + ( √99 - √99) + √100 =
-1 + 0 + 0 +........+ 0 + √100 =
-1 + √100 =
-1 + 10 =
9
\(S\ =\ \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{1}{\sqrt{n}+\sqrt{n+1}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{1}{\sqrt{n}+\sqrt{n+1}}\cdot \frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n}-\sqrt{n+1}}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{\sqrt{n}-\sqrt{n+1}}{(n)-(n+1)}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\frac{\sqrt{n}-\sqrt{n+1}}{-1}\\~\\ S\ =\ \sum\limits_{n=1}^{99}\sqrt{n+1}-\sqrt{n}\\~\\ S\ =\ {\color{}\sqrt2}-\sqrt1\ {\color{}+\sqrt3-\sqrt2+\sqrt4-\sqrt3+\sqrt5-\sqrt4+\dots+\sqrt{99}-\sqrt{98}}+\sqrt{100}\ {\color{}-\ \sqrt{99}}\\~\\ S\ =\ {\color{gray}\sqrt2}-\sqrt1\ {\color{gray}+\sqrt3-\sqrt2+\sqrt4-\sqrt3+\sqrt5-\sqrt4+\dots+\sqrt{99}-\sqrt{98}}+\sqrt{100}\ {\color{gray}-\ \sqrt{99}}\\~\\ S\ =\ -\sqrt1+\sqrt{100}\\~\\ S\ =\ -1+10\\~\\ S\ =\ 9 \)_