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Question Number 54641 by Joel578 last updated on 08/Feb/19

$$\mathrm{Given}$$$$f\left(x\right)=\frac{4x+\sqrt{4x^2-1}}{\sqrt{2x+1}-\sqrt{2x-1}}$$$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}$$$$f\left(13\right)+f\left(14\right)+f\left(15\right)+...+f\left(112\right)$$

Commented by Meritguide1234 last updated on 08/Feb/19

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{(2\mathrm{x}-1)+(2\mathrm{x}+1)+\sqrt{(2\mathrm{x}+1)(2\mathrm{x}-1)}}{\sqrt{2\mathrm{x}+1}+\sqrt{2\mathrm{x}-1}}$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\mathrm{denominator}\mathrm{should}\mathrm{be}\mathrm{positive}$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{{\left(\sqrt{2\mathrm{x}-1}\right)}^3-{\left(\sqrt{2\mathrm{x}+1}\right)}^3}{2}$$$$\mathrm{f}\left(13\right)+\mathrm{f}\left(14\right)+...+\mathrm{f}\left(112\right)=1625$$

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