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
Expand $(1 + x)^{-1}$ up to and including the term in $x^3$ , and state the range of values of $x$ for which the expansion is valid.
Level 3 · 3 marksTry it
2
Expand $(1 + 2x)^{-3}$ in ascending powers of $x$ , up to and including the term in $x^{3}$ , and state the range of $x$ for which the expansion is valid. (4 marks)
Level 3 · 4 marksTry it
3
(a) Expand $(9 - 3x)^{\frac{1}{2}}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ , and state the range of validity. (4 marks)
(b) By substituting $x = 0.3$ , find an approximation to $\sqrt{8.1}$ , giving your answer to $5$ decimal places. (2 marks)
Level 3 · 6 marksTry it
4
(a) Expand $(1 - 4x)^{\frac{1}{2}}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ . (3 marks)
(b) Explain why the expansion cannot be used to estimate $\sqrt{1 - 4(0.3)}$ . (1 mark)
(c) Use the expansion with a suitable value of $x$ to estimate $\sqrt{0.92}$ , giving your answer to $4$ decimal places. (2 marks)
Level 3 · 6 marksTry it
5
Expand $\dfrac{1}{2 + x}$ in ascending powers of $x$ , up to and including the term in $x^{3}$ , and state the range of validity. (4 marks)
Level 3 · 4 marksTry it

5 more questions on binomial expansion for any index

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