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2 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2 · Question 3(c)(i)

By substituting suitable values of x on both sides of the expansion of (1 + x)^n = sum_{r=0}^n binom(n, r) x^r:

Show that sum_{r=0}^n binom(n, r) = 2^n.

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Other parts of this question

  1. 3(a)(i)Use the fact that 1/r - 1/(r + 1) = 1 / (r(r + 1)) to show that S_n = sum_{r=1}^n 1/(r(r + 1)) = 1 - 1/(n + 1).[5 marks]
  2. 3(a)(ii)Deduce that as n -> infinity, S_n -> 1.[1 mark]
  3. 3(b)The common ratio, r, of a geometric series is given by r = 5x / (4 + x^2). Find ALL the values of x for which the series converges.[10 marks]
  4. 3(c)(ii)Show that sum_{r=0}^n binom(n, r) (-1)^r = 0.[2 marks]

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