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.

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

1
(a) Find the first three terms, in ascending powers of $x$ , of the binomial expansion of

$\sqrt{9 - 8x}$

writing each term in simplest form.

(3 marks)

A student substitutes $x = \dfrac{1}{4}$ into these three terms to estimate the value of $\sqrt{7}$
(b) With reference to the terms of your expansion, and without doing any calculations, explain whether the student's estimate is an overestimate or an underestimate of $\sqrt{7}$ .

(1 mark)
Level 2 · 4 marksTry it
2
(a) Find the first three terms, in ascending powers of $x$ , of the binomial expansion of

$(8 - 3x)^{\frac{1}{3}}$

writing each term in simplest form.

(3 marks)
(b) Explain why the expansion cannot be used with $x = -3$ to estimate $\sqrt[3]{17}$

(1 mark)
Level 2 · 4 marksTry it
3
(a) Find the first three terms, in ascending powers of $x$ , of the binomial expansion of

$(8+3x)^{\frac{2}{3}}$ ,

and state the range of values of $x$ for which the expansion is valid. (4 marks)
(b) Use your expansion to estimate $8.3^{\frac{2}{3}}$ , giving your answer to 4 decimal places. (1 mark)
Level 2 · 5 marksTry it
4
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
5
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

28 more questions on binomial expansion for any index

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