Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series

P1^

3

22 The 1st term of an arithmetic progression is a and the common difference is

d, where d ≠ 0.

(i) Write down expressions, in terms of a and d, for the 5th term and the

15th term.

The 1st term, the 5th term and the 15th term of the arithmetic progression

are the first three terms of a geometric progression.

(ii) Show that 3a = 8d.

(iii) Find the common ratio of the geometric progression.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q4 November 2007]

INvESTIGaTIONS

Snowflakes

Draw an equilateral triangle with sides 9 cm long.

Trisect each side and construct equilateral triangles on the middle section of each

side as shown in diagram (b).

Repeat the procedure for each of the small triangles as shown in (c) and (d) so that

you have the first four stages in an infinite sequence.

Calculate the length of the perimeter of the figure for each of the first six steps,

starting with the original equilateral triangle.

What happens to the length of the perimeter as the number of steps increases?

Does the area of the figure increase without limit?

Achilles and the tortoise

Achilles (it is said) once had a race with a tortoise. The tortoise started 100 m

ahead of Achilles and moved at 101 ms–1 compared to Achilles’ speed of 10 ms–1.

Achilles ran to where the tortoise started only to see that it had moved 1 m fur-

ther on. So he ran on to that spot but again the tortoise had moved further on,

this time by 0.01 m. This happened again and again: whenever Achilles got to the

spot where the tortoise was, it had moved on. Did Achilles ever manage to catch

the tortoise?

(a) (b) (c) (d)

Figure 3.4