Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series

P1^

3

10 (i) Find the first three terms, in descending powers of x, in the expansion

of (^) ()x−x^2
5
.
(ii) Hence find the coefficient of x in the expansion of (^4122)
5
()+ ()−
x
x
x

.

11 (i) Show that (2 + x)^4 = 16 + 32 x + 24 x^2 + 8 x^3 + x^4 for all x.

(ii) Find the values of x for which (2 + x)^4 = 16 + 16 x + x^4.

[MEI]

12 The first three terms in the expansion of (2 + ax)n, in ascending powers of x,

are 32 − 40 x + bx^2. Find the values of the constants n, a and b.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q4 June 2006]

13 (i) Find the first three terms in the expansion of (2 – x)^6 in ascending

powers of x.

(ii) Find the value of k for which there is no term in x^2 in the expansion of

(1 + kx)(2 − x)^6.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q4 June 2005]

14 (i) Find the first three terms in the expansion of (1 + ax)^5 in ascending

powers of x.

(ii) Given that there is no term in x in the expansion of (1 − 2 x)(1 + ax)^5 ,

find the value of the constant a.

(iii) For this value of a, find the coefficient of x^2 in the expansion of (1 − 2 x)

(1 + ax)^5.

[Cambridge AS & A Level Mathematics 9709, Paper 12 Q6 June 2010]

INvESTIGaTIONS

Routes to victory

In a recent soccer match, Juventus beat Manchester United 2–1.

What could the half-time score have been?

(i) How many different possible half-time scores are there if the final score is

2–1? How many if the final score is 4–3?

(ii) How many different ‘routes’ are there to any final score? For example, for the

above match, putting Juventus’ score first, the sequence could be:

0–0 → 0–1 → 1–1 → 2–1

or 0–0 → 1–0 → 1–1 → 2–1

or 0–0 → 1–0 → 2–0 → 2–1.

So in this case there are three routes.

Investigate the number of routes that exist to any final score (up to a maximum

of five goals for either team).

Draw up a table of your results. Is there a pattern?