N2014 P1 Q6

N2014 P1 Q6

Junior College 2
12 marks
  1. A sequence \(p_1,p_2,p_3,\ldots\) is defined by \(p_1=1\) and \(p_{n+1}=4p_n-7\), for \(n\ge1\).
    1. Prove by mathematical induction that \(p_n=\frac13(7-4^n)\).[5]
    2. Find an expression for \(\displaystyle\sum_{r=1}^n p_r\).[3]
  2. The sum of the first \(n\) terms of a sequence \(u_1,u_2,u_3,\ldots\) is \(S_n=1-\frac1{(n+1)!}\).
    1. Explain why the series is convergent and state its sum to infinity.[2]
    2. Find an expression for \(u_n\).[2]

Solution:

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Answer:(a)(i) \(p_n=(7-4^n)/3\) (proven). (ii) \(\frac{7n}{3}-\frac{4(4^n-1)}9\). (b)(i) \(1\). (ii) \(u_n=\frac n{(n+1)!}\).

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