A-level · Sequences & series

The binomial expansionExam-style questions

Expanding (a + b)^n with binomial coefficients — Pascal's triangle and nCr — and picking out a single term or coefficient without expanding everything.

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

DifficultyAll Level 2 Level 3 Level 4

1
(a) Find the first four terms in the expansion of $(1 - 2x)^7$ in ascending powers of $x$ .

(4 marks)

In the expansion of $(1 - 2y - 4y^2)^7$ in ascending powers of $y$ , the coefficient of $y^3$ is $c$ .
(b) By writing $1 - 2y - 4y^2$ in the form $1 - 2x$ , use your answer to part (a) to determine the value of $c$ .

(4 marks)
Level 4 · 8 marksTry it
2
(a) Find the first 4 terms, in ascending powers of $x$ , of the binomial expansion of

$(2 + kx)^5$

where $k$ is a non-zero constant. Give each term in simplest form.

(4 marks)

The coefficient of $x$ and the coefficient of $x^2$ in this expansion are, respectively, the first term and the second term of an arithmetic series.

The sum of the first 5 terms of this arithmetic series is $1200$
(b) Use algebra to find each possible value of $k$ .

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)
Level 4 · 7 marksTry it
3
$\text{g}(x) = (2 + px)^9$

where $p$ is a negative constant.
(a) Write down the constant term in the binomial expansion of $\text{g}(x)$ .

(1 mark)

In the binomial expansion of $\text{g}(x)$ , the coefficient of $x^2$ is four times the constant term.
(b) (i) Find the value of $p$ .

(ii) Hence find the coefficient of $x$ in the binomial expansion of $\text{g}(x)$ .

(6 marks)
Level 4 · 7 marksTry it
4
(a) Expand

$(4 + ax)^5$

where $a$ is a positive constant, in ascending powers of $x$ up to and including the term in $x^2$ . Give each term in simplest form.

(3 marks)

$\text{h}(x) = \left(1 - \frac{3}{x}\right)(4 + ax)^5$

When $\text{h}(x)$ is expanded in ascending powers of $x$ , the term in $x$ is $-5120x$ .
(b) Use your answer to part (a) to find the value of $a$ .

(4 marks)
Level 4 · 7 marksTry it
5
The coefficient of $x^{3}$ in the expansion of $(2 + ax)^{7}$ is $-70$ .
(a) Find the value of the constant $a$ .

(2 marks)
(b) Hence find the coefficient of $x^{3}$ in the expansion of $(3 + 4x)(2 + ax)^{7}$ .

(3 marks)
Level 4 · 5 marksTry it