Recurrence Relations with Given Sum Condition

A question is this type if and only if it defines a sequence by a recurrence relation and uses a condition on the sum of the first few terms to find constants or specific term values.

2 questions · Standard +0.6

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Edexcel P2 2021 January Q8
7 marks Standard +0.8
8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { n + 1 } = 2 \left( a _ { n } + 3 \right) ^ { 2 } - 7 \\ a _ { 1 } = p - 3 \end{gathered}$$ where \(p\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(p\), giving your answer in simplest form. Given that \(\sum _ { n = 1 } ^ { 3 } a _ { n } = p + 15\)
  2. find the possible values of \(a _ { 2 }\)
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P2 2023 June Q11
8 marks Standard +0.3
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n } \\ u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.
  1. Find, in terms of \(a\) and \(b\),
    1. \(u _ { 2 }\)
    2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
  2. \(b = a + 9\)
  3. show that
    4. $$a ^ { 2 } - 5 a - 66 = 0$$
  4. Hence find the larger possible value of \(u _ { 2 }\)