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4 marksSequences

CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2 · Question 3(a)

A sequence is defined by the recurrence relation u_(n+1) = u_(n-1) + x(u_n)', where u_1 = 1, u_2 = x and (u_n)' is the derivative of u_n. For example, u_3 = 1 + x.

Given that u_8 = 13x + 1 and that u_10 = 34x + 1, find (u_9)'.

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Other parts of this question

  1. 3(b)(i)Show that S_n = (n(n^2 - 1)) / 3.[7 marks]
  2. 3(b)(ii)Hence, or otherwise, evaluate sum_{r=10}^20 r(r - 1).[5 marks]
  3. 3(c)(i)Given that ^n P_r = n! / (n - r)!, show that (^2r P_r * ^n P_r) / ((2r)!) is equal to the binomial coefficient ^n C_r.[4 marks]
  4. 3(c)(ii)Determine the coefficient of the term in x^3 in the binomial expansion of (3x + 2)^5.[5 marks]

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