- #1
sooyong94
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Homework Statement
Someone please check my work... :D
If ##f(x)=\sqrt{x}+\sqrt{x+1}## , find the value of
##\frac{1}{f(1)}+\frac{1}{f(2)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}##
Homework Equations
Summation of series, rationalizing the denominator.
The Attempt at a Solution
##f(x)=\sqrt{x}+\sqrt{x+1}##
##f(x)=\sqrt{x}+\sqrt{x+1}(\frac{\sqrt{x}-\sqrt{x+1}}{\sqrt{x}-\sqrt{x+1}})##
##f(x)=\frac{x-(x+1)}{\sqrt{x}-\sqrt{x+1}}##
##\frac{1}{f(x)}=\sqrt{x+1}-\sqrt{x}##
##\frac{1}{f(1)}+\frac{1}{f(2)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}##
##=\sum_{r=1}^{24} \sqrt{x+1}-\sqrt{x}##
##=(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+...+(\sqrt{24}-\sqrt{23})+(\sqrt{25}-\sqrt{24})##
##=\sqrt{25}-\sqrt{1}##
##=4##
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