A-level · Sequences & series

Binomial expansion for any indexExam-style questions

Expanding (1 + x)^n and (a + bx)^n when n is negative or fractional — the series form, when it's valid, and combining it with partial fractions.

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33 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    (a) Find the binomial expansion of

    9−x\sqrt{9 - x}

    in ascending powers of xx, up to and including the term in x3x^3. Give each coefficient as a simplified fraction.

    (4 marks)
    (b) State the range of values of xx for which this expansion is valid.

    (1 mark)
    (c) Hence find the expansion of (18+x)9−x(18 + x)\sqrt{9 - x} in ascending powers of xx, up to and including the term in x3x^3

    (2 marks)
    (d) Use your answer to part (c) with x=1x = 1 to find a rational approximation to 2\sqrt{2}, giving your answer as a fraction in its simplest form. You must show each step of your working.

    (3 marks)
    Level 5 · 10 marksTry it
  2. 2
    Find the first three terms, in ascending powers of xx, of (4+x)1/2(4 + x)^{1/2}, and state the range of validity.
    Level 5 · 4 marksTry it
  3. 3
    Given that 3(1+x)(1−2x)≡11+x+21−2x\dfrac{3}{(1 + x)(1 - 2x)} \equiv \dfrac{1}{1 + x} + \dfrac{2}{1 - 2x}, find the expansion of 3(1+x)(1−2x)\dfrac{3}{(1 + x)(1 - 2x)} up to and including the term in x2x^2, and state the range of validity.
    Level 5 · 4 marksTry it
  4. 4
    f(x)=7+x(1+x)(2−x)\mathrm{f}(x) = \frac{7 + x}{(1 + x)(2 - x)}
    (a) Express f(x)\mathrm{f}(x) in partial fractions. (3 marks)
    (b) Hence find the first three terms, in ascending powers of xx, of the expansion of f(x)\mathrm{f}(x). (4 marks)
    (c) State the range of xx for which the full expansion is valid. (1 mark)
    Level 5 · 8 marksTry it
  5. 5
    (a) Expand (1−x)12(1 - x)^{\frac{1}{2}} in ascending powers of xx, up to and including the term in x3x^{3}. (3 marks)
    (b) Substitute x=150x = \dfrac{1}{50} to evaluate the expansion, giving 88 decimal places. (2 marks)
    (c) Using 1−150=752\sqrt{1 - \dfrac{1}{50}} = \dfrac{7}{5\sqrt{2}}, deduce an approximation to 2\sqrt{2}, giving your answer to 55 decimal places. (2 marks)
    Level 5 · 7 marksTry it

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