Consider the sequence defined by ${{u}_{n+1}}=\frac{2}{{{u}_{n}}-2}+3$, for all positive integers of

Question

Consider the sequence defined by ${{u}_{n+1}}=\frac{2}{{{u}_{n}}-2}+3$, for all positive integers of $n$, and ${{u}_{0}}=3$. Find the values of ${{u}_{1}}$, ${{u}_{2}}$ and ${{u}_{3}}$. [2] Suppose ${{u}_{n}}$ converges to $l$ as $n	o \infty $. By forming and solving a suitable quadratic equation, find the value of $l$. [4] Under what condition will ${{u}_{n}}$ converge to the other limit you rejected in (ii)? You should use calculations obtained from your G.C. to substantiate your answer. [1]

Accepted Answer

(i) $u_1=5, u_2=\frac{11}{3}, u_3=\frac{21}{5}$. (ii) $l=4$. (iii) $u_0=1$, giving the constant sequence $1,1,1,\ldots$.

2007 SAJC P1 Q10

2007 SAJC P1 Q10

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