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Tim Gan Math

2012 A-Level H2 Mathematics

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Question 3Recurrence Relations and Summation

A sequence \(u_1, u_2, u_3, \dots\) is given by

\[u_1 = 2 \text{ and}\hspace{0.5em} u_n = \frac{3u_{n-1} - 1}{6} \text{ for}\hspace{0.5em} n \ge 2.\]

  1. Find the exact values of \(u_2\) and \(u_3\).
  2. It is given that \(u_n \to l\) as \(n \to \infty\). Showing your working, find the exact value of \(l\).
  3. 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.
Solution step 1, image 1
Solution step 2, image 1
Solution step 3, image 1
Solution step 3, image 2

Final Answer:

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