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Given-f-x-4x-4x-2-1-2x-1-2x-1-Find-the-value-of-f-13-f-14-f-15-f-112-




Question Number 54641 by Joel578 last updated on 08/Feb/19
Given  f(x) = ((4x + (√(4x^2  − 1)))/( (√(2x + 1)) − (√(2x − 1))))  Find the value of   f(13) + f(14) + f(15) + ... + f(112)
$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{4}{x}\:+\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\:\mathrm{1}}}{\:\sqrt{\mathrm{2}{x}\:+\:\mathrm{1}}\:−\:\sqrt{\mathrm{2}{x}\:−\:\mathrm{1}}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$${f}\left(\mathrm{13}\right)\:+\:{f}\left(\mathrm{14}\right)\:+\:{f}\left(\mathrm{15}\right)\:+\:…\:+\:{f}\left(\mathrm{112}\right) \\ $$
Commented by Meritguide1234 last updated on 08/Feb/19
f(x)=(((2x−1)+(2x+1)+(√((2x+1)(2x−1))))/( (√(2x+1))+(√(2x−1))))  ⇒f(x)=denominator should be positive  ⇒f(x)=((((√(2x−1)))^3 −((√(2x+1)))^3 )/2)  f(13)+f(14)+...+f(112)=1625
$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\left(\mathrm{2x}−\mathrm{1}\right)+\left(\mathrm{2x}+\mathrm{1}\right)+\sqrt{\left(\mathrm{2x}+\mathrm{1}\right)\left(\mathrm{2x}−\mathrm{1}\right)}}{\:\sqrt{\mathrm{2x}+\mathrm{1}}+\sqrt{\mathrm{2x}−\mathrm{1}}} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{denominator}\:\mathrm{should}\:\mathrm{be}\:\mathrm{positive} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\frac{\left(\sqrt{\mathrm{2x}−\mathrm{1}}\right)^{\mathrm{3}} −\left(\sqrt{\mathrm{2x}+\mathrm{1}}\right)^{\mathrm{3}} }{\mathrm{2}} \\ $$$$\mathrm{f}\left(\mathrm{13}\right)+\mathrm{f}\left(\mathrm{14}\right)+…+\mathrm{f}\left(\mathrm{112}\right)=\mathrm{1625} \\ $$

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