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6 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2 · Question 4(b)

Show that \frac{n!}{(n - r)!r!} + \frac{n!}{(n - r + 1)!(r - 1)!} = \frac{(n + 1)!}{(n - r + 1)!r!}.

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Other parts of this question

  1. 4(a)(i)For the binomial expansion of (2x + 3)^{20}, show that the ratio of the term in x^6 to the term in x^7 is \frac{3}{4x}.[5 marks]
  2. 4(a)(ii)a)Determine the FIRST THREE terms of the binomial expansion of (1 + 2x)^{10}.[4 marks]
  3. 4(a)(ii)b)Hence, obtain an estimate for (1.01)^{10}.[3 marks]
  4. 4(c)(i)Show that the function f(x) = -x^3 + 3x + 4 has a root in the interval [1, 3].[3 marks]
  5. 4(c)(ii)By taking x_1 = 2.1 as a first approximation of the root in the interval [1, 3], use the Newton–Raphson method to obtain a second approximation, x_2, in…[4 marks]

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