A sequence $u_1, u_2, u_3, \dots$ is given by \[u_1 = 2 \text{ and}\hspace{0.5em} u_n = \frac{3u_{

Question

A sequence $u_1, u_2, u_3, \dots$ is given by $$u_1 = 2 \text{ and}\hspace{0.5em} u_n = \frac{3u_{n-1} - 1}{6} \text{ for}\hspace{0.5em} n \ge 2.$$ Find the exact values of $u_2$ and $u_3$. It is given that $u_n \to l$ as $n \to \infty$. Showing your working, find the exact value of $l$. For this value of $l$, show that $u_n = \frac{14}{3}\left(\frac{1}{2}\right)^n + l$ satisfies both the initial condition and the recurrence relation given in the question.

Accepted Answer

(i) $u_2=\dfrac56$, $u_3=\dfrac14$ (ii) $l=-\dfrac13$ (iii) $u_n=\dfrac{14}{3}\left(\dfrac12\right)^n-\dfrac13$ satisfies the initial condition and recurrence.

N2012 P1 Q3 [Modified]

N2012 P1 Q3 [Modified]

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