A sequence of negative numbers is defined by ${{x}_{n\,+1}}=\frac{2-3{{x}_{n}}}{{{x}_{n}}-4}$, whe

Question

A sequence of negative numbers is defined by ${{x}_{n\,+1}}=\frac{2-3{{x}_{n}}}{{{x}_{n}}-4}$, where ${{x}_{1}}=-\frac{1}{7}$. Write down the values of ${{x}_{2}}$ and ${{x}_{3}}$, giving your answers correct to $3$ significant figures.[2] Given that as $n\to \infty $, ${{x}_{n}}\to \ell $, find, without the use of a graphic calculator, the value of $\ell $.[3]

Accepted Answer

(i) ${{x}_{2}}=-0.586$ and ${{x}_{3}}=-0.820$ (ii) $\ell =-1$

2007 RJC P1 Q3

2007 RJC P1 Q3

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