Key pointsP1^
3
Cubes
A cube is painted red. It is then cut up into a
number of identical cubes, as in figure 3.5.
How many of the cubes have the following
numbers of faces painted red?
(i) 3 (ii) 2 (iii) 1 (iv) 0
In figure 3.5 there are 125 cubes but your
answer should cover all possible cases. Figure 3.5KEy POINTS
1 A sequence is an ordered set of numbers, u 1 , u 2 , u 3 , ..., uk, ... un, where uk
is the general term.
2 In an arithmetic sequence, uk+1 = uk + d where d is a fixed number called
the common difference.
3 In a geometric sequence, uk+1 = ruk where r is a fixed number called the
common ratio.
4 For an arithmetic progression with first term a, common difference d and n
terms:
● the kth term uk = a + (k − 1)d
● the last term l = a + (n − 1)d
●● the sum of the terms = 12 na()+=ln 21 [ 21 an+(– )d].
5 For a geometric progression with first term a, common ratio r and n terms:
● the kth term uk = ark–1
● the last term an = arn–1
● the sum of the terms = ar
rar
r(–n ) n
(– )(– )
(–)
1
1
1
1
=.
6 For an infinite geometric series to converge, − 1 r 1.
In this case the sum of all the terms is given by (– 1 ar).7 Binomial coefficients, denoted by
n
r
or
nC
r, can be found
● using Pascal’s triangle
● using tables
● using the formula n
rn
rn r
= −
()
!
!!.
8 The binomial expansion of (1 + x)n may also be written() (– )
!(– )( –)
!
11 1 –
2
12
3
+=xnn ++x nn x^2 +nn n xn^3 +...+ xn^1 +xxn.