Expansion of (a+bx^m)^n

Expand expressions where the variable appears as x^m (m>1) rather than x, such as (1+x²)^n or (8+27x³)^(1/3).

3 questions · Standard +0.6

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CAIE P3 2003 November Q2
4 marks Moderate -0.8
2 Expand \(\left( 2 + x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
AQA Paper 2 2019 June Q9
9 marks Standard +0.3
  1. Show that the first two terms of the binomial expansion of \(\sqrt{4 - 2x^2}\) are $$2 - \frac{x^2}{2}$$ [2 marks]
  2. State the range of values of \(x\) for which the expansion found in part (a) is valid. [2 marks]
  3. Hence, find an approximation for $$\int_0^{0.4} \sqrt{\cos x} \, dx$$ giving your answer to five decimal places. Fully justify your answer. [4 marks]
  4. A student decides to use this method to find an approximation for $$\int_0^{1.4} \sqrt{\cos x} \, dx$$ Explain why this may not be a suitable method. [1 mark]
Edexcel AEA 2014 June Q4
13 marks Hard +2.3
Given that $$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$
  1. show that $$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$ [5]
  2. show that \((9 - 4x^2)^{-\frac{1}{2}}\) can be written in the form \(\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}\) and give \(q\) in terms of \(r\). [3]
  3. Find \(\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}\) [3]
  4. Hence find the exact value of $$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$ giving your answer as a rational number. [2]