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Question Number 202104 by necx122 last updated on 20/Dec/23

Commented by necx122 last updated on 20/Dec/23

$p l e a s e h e l p w i t h t h i s$
Commented by cortano12 last updated on 21/Dec/23

$S = \underset{i = 1}{\sum^{99}} (\frac{1}{\sqrt{i} + \sqrt{i + 1}}) = \underset{i = 1}{\sum^{99}} (\frac{\sqrt{i} - \sqrt{i + 1}}{- 1})$ $(telescopic series)$ $S = - (\sqrt{1} - \sqrt{100}) = - (- 9) = 9$
	Answered by esmaeil last updated on 20/Dec/23

	$\frac{\sqrt{1} - \sqrt{2}}{- 1} + \frac{\sqrt{2} - \sqrt{3}}{- 1} + \frac{\sqrt{3} - \sqrt{4}}{- 1} + . . . + \frac{\sqrt{99} - \sqrt{100}}{- 1} =$ $= 9$
	Commented by necx122 last updated on 20/Dec/23
	$Sincerely, I still do not understand. At forst I thought the conjugate of each fraction was taken and that′s how it was done for all but not at all. Please shed some more light anyone$
	$S i n c e r e l y, I s t i l l d o n o t u n d e r s t a n d . A t f o r s t$ $I t h o u g h t t h e c o n j u g a t e o f e a c h f r a c t i o n$ $w a s t a k e n a n d {t h a t}^{'} s h o w i t w a s d o n e$ $f o r a l l b u t n o t a t a l l . P l e a s e s h e d s o m e$ $m o r e l i g h t a n y o n e$
	Commented by esmaeil last updated on 20/Dec/23

	$\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + . . . + \frac{1}{\sqrt{99} + \sqrt{100}} = A$ $\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{1}{\sqrt{a} + \sqrt{b}} \times \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b} ★$ $\frac{1}{\sqrt{1} + \sqrt{2}} \overset{★}{=} \frac{\sqrt{1} - \sqrt{2}}{1 - 2 = - 1}, \frac{1}{\sqrt{2} + \sqrt{3}} \overset{★}{=} \frac{\sqrt{2} - \sqrt{3}}{2 - 3 = - 1},$ $. . ., \frac{1}{\sqrt{99} + \sqrt{100}} \overset{★}{=} \frac{\sqrt{99} - \sqrt{100}}{99 - 100 = - 1}$ $\Rightarrow$ $A = \frac{\sqrt{1} - \sqrt{2} + \sqrt{2} - \sqrt{3} + \sqrt{3} - \sqrt{4} . . . - \sqrt{100}}{- 1}$ $= \frac{1 - \sqrt{100}}{- 1} = 9$
	Commented by necx122 last updated on 21/Dec/23
	Thanks so much.
	Answered by MM42 last updated on 20/Dec/23

	$s = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + . . . + (\sqrt{99} - \sqrt{98}) + (\sqrt{100} - \sqrt{99})$ $= \sqrt{100} - \sqrt{1} = 9 ✓$
	Commented by MathematicalUser2357 last updated on 22/Dec/23

	$How did you get ✓$
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