2025 NYJC Promo Q4

2025 NYJC Promo Q4

Junior College 1
7 marks
  1. A sequence \(u_1, u_2, ...\) is such that \(\sum_{r=1}^{n} u_r = \frac{3n+26}{4n-2}\).
    1. Show that the series \(\sum_{r=1}^{\infty} u_r\) converges and write down the value of the sum to infinity.[2]
    2. Find the exact value of \(\sum_{r=18}^{\infty} u_r\).[2]
  2. A geometric sequence \(a_1, a_2, ...\) is such that \(a_{n+1}=pa_n+q\), where \(p\) and \(q\) are constants.
    1. State the value of \(q\).[1]
    2. Given that \(a_6=\frac{32}{243}a_1\), find \(p\).[2]

Solution:

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Answer:(a)(i) \(\dfrac34\) (a)(ii) \(-\dfrac5{12}\) (b)(i) \(q=0\) (b)(ii) \(p=\dfrac23\)

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