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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Find the first three terms in the expansion of $(4 - 3x)^{-\frac{1}{2}}$ in ascending powers of $x$ .

(4 marks)
(b) State the range of values of $x$ for which the expansion in part (a) is valid.

(1 mark)
(c) Find the coefficient of $x^2$ in the expansion of $\dfrac{2 + 5x}{\sqrt{4 - 3x}}$ .

(4 marks)
Level 4 · 9 marksTry it
2
(a) Find the first three terms in the expansion of $(4 - 16x)^{\frac{1}{2}}$ in ascending powers of $x$ .

(4 marks)
(b) State the range of values of $x$ for which the expansion in part (a) is valid.

(1 mark)
(c) It is given that the coefficient of $x^2$ in the expansion of $(1 + ax)^2(4 - 16x)^{\frac{1}{2}}$ is $6$ .

Determine the possible values of the constant $a$ .

(4 marks)
Level 4 · 9 marksTry it
3
(a) Find the first three terms in the expansion of $(9 + 2x)^{-\frac{1}{2}}$ in ascending powers of $x$ .

(4 marks)
(b) State the set of values of $x$ for which this expansion is valid.

(1 mark)
(c) For small values of $x$ , $\dfrac{6 + ax}{\sqrt{9 + 2x}} \approx 2 + kx^2$ , where $a$ and $k$ are constants.

Determine the value of the constant $a$ .

(2 marks)
Level 4 · 7 marksTry it
4
The diagram shows triangle $PQR$ , in which $PQ = (8 - 2h)$ cm, $PR = (3 + h)$ cm and angle $QPR = 60°$ , where $0 < h < 4$ .(a) Show that $QR^2 = 49 - 28h + 7h^2$ .

(2 marks)
(b) Given that $h$ is small, use a binomial expansion to find the values of the constants $m$ and $n$ for which $QR \approx 7 + mh + nh^2$ .

(4 marks)
Level 4 · 6 marksTry it
5
In this question you should assume that $-\tfrac{1}{2} < x < \tfrac{1}{2}$ .
(a) (i) Find the first four terms in the expansion of $(1 + 2x)^{-2}$ in ascending powers of $x$ .

(ii) Find, in terms of $n$ , the coefficient of $x^n$ in this expansion, where $n$ is a positive integer.

(3 marks)
(b) Express $\displaystyle\sum_{r=0}^{\infty} (-2x)^r$ as a single fraction.

(1 mark)
(c) Hence, or otherwise, express $\displaystyle\sum_{r=0}^{\infty} (r + 3)(-2x)^r$ in the form $\dfrac{a + bx}{(1 + 2x)^2}$ , where $a$ and $b$ are integers to be determined.

(3 marks)
Level 4 · 7 marksTry it

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