# OCR A Level: Further Pure 2 - Iteration [June 2008 Q4]

Algebra Level pending

$$(\text{i})$$ Sketch, on the same diagram, $$y=\text{sech} x$$ and $$y=x^2$$.

$$(\text{ii})$$ Use the definition of $$\text{sech} x$$ in terms of $$e^x$$ and $$e^{-x}$$ to show the $$x$$-coordinates of the points of intersection are solutions to the equation

$x^2= \dfrac{2e^x}{e^{2x}+1}.$

$$(\text{iii})$$ The iteration $${ x }_{ n+1 }^{ 2 }= \dfrac{2e^{x_n}}{e^{2 x_n }+1}$$ can be used to find the positive root. With initial value $$x_1=1$$, we obtain $$x_2 = 0.8050$$, $$x_3 = 0.8633$$, $$x_4 = 0.8463$$ and $$x_5 = 0.8513$$, correct to 4 decimal places. State, with a reason, whether this iteration produces a "staircase" or a "cobweb" diagram.

Input 1 if the iteration forms a "staircase", or 0 if it forms a "cobweb".

###### This is part of the set OCR A Level Problems.
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