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CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2 · Question 3(b)

A GP with first term a and common ratio r has sum to infinity 81 and the sum of the first four terms is 65. Find the values of a and r.

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  1. 3(a)(i)A sequence \{u_n\} is defined by the recurrence relation u_{n+1} = u_n + n, u_1 = 3, n \in \mathbb{N}. State the first FOUR terms of the sequence.[3 marks]
  2. 3(a)(ii)Prove by mathematical induction, or otherwise, that u_n = \frac{n^2 - n + 6}{2}.[8 marks]
  3. 3(c)(i)Write down the first FIVE terms in the power series expansion of \ln(1 + x), stating the range of values of x for which the series is valid.[3 marks]
  4. 3(c)(ii)a)Using the result from (c)(i) above, obtain a similar expansion for \ln(1 - x).[2 marks]
  5. 3(c)(ii)b)Hence, prove that \ln\left(\frac{1 + x}{1 - x}\right) = 2\left(x + \frac{1}{3}x^3 + \frac{1}{5}x^5 + \dots\right).[3 marks]

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