2a=(1/( (√1)+(√2)))+(1/( (√2)+(√3)))+...+(1/( (√(2024))+(√(2025))))=?


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Question Number 163325 by mathlove last updated on 06/Jan/22

$2 a = \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + . . . + \frac{1}{\sqrt{2024} + \sqrt{2025}} = ?$
Commented by cortano1 last updated on 06/Jan/22

$2 a = \underset{k = 1}{\sum^{2024}} \frac{1}{\sqrt{k} + \sqrt{k + 1}} = \underset{k = 1}{\sum^{2024}} (\sqrt{k + 1} - \sqrt{k})$ $2 a = \sqrt{2025} - \sqrt{1}$ $2 a = 45 - \sqrt{1}$
Commented by blackmamba last updated on 06/Jan/22

$t e l e s c o p i c$
	Answered by mr W last updated on 06/Jan/22

	$2 a = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + . . . + (\sqrt{2025} - \sqrt{2024})$ $2 a = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + . . . + (\sqrt{2025} - \sqrt{2024})$ $2 a = - \sqrt{1} + \sqrt{2025}$ $2 a = - 1 + 45 = 44$
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