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Mathematics Term 1 STPM Chapter 2 Sequences and Series
The binomial theorem

When a binomial expression such as (a + b) is raised to the power of n (where n is a small positive integer),
we obtain a binomial expansion such as follows:
0
When n = 0, (a + b) = 1
n = 1, (a + b) = a + b
1
2
2
n = 2, (a + b) = a + 2ab + b 2
n = 3, (a + b) = a + 3a b + 3ab + b 3
2
2
3
3
2 2
4
3
4
3
n = 4, (a + b) = a + 4a b + 6a b + 4ab + b 4
5
n = 5, (a + b) = a + 5a b + 10a b + 10a b + 5ab + b 5
4
3 2
5
2 3
4
The coefficients of each binomial expansion form an array which is known as a Pascal triangle, as shown
2 below:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each number in the Pascal triangle is the sum of two numbers adjacent to it in the previous line. For example,
10 = 4 + 6. This means that the triangle can be extended to obtain the coefficients of the binomial expansion
n
of (a + b) for higher values of n. This way of expansion is rather tedious. An alternative method, known as
the binomial theorem, can be used to expand (a + b) for any n  Z .
n
+
Notice that we can arrange the above binomial expansions in the following way:
(a + b) = a + 3a b + 3 · 2 ab + b 3
3
2
3
2
1 · 2
3
4
4
3
2 2
(a + b) = a + 4a b + 4 · 3 a b + 4 · 3 · 2 ab + b 4
1 · 2 1 · 2 · 3
5
3 2
4
5
(a + b) = a + 5a b + 5 · 4 a b + 5 · 4 · 3 a b + 5 · 4 · 3 · 2 ab + b 5
4
2 3
1 · 2 1 · 2 · 3 1 · 2 · 3 · 4
In general, if n is a positive integer, we have
n – 2 2
n
3
(a + b) = a + na n – 1 b + n(n – 1) a b + n(n – 1)(n – 2) a n – 3 b + … + b .
n
n
1 · 2 1 · 2 · 3
n
n
n
However, since n = 1 2 , n(n – 1) = 1 2 , n(n – 1)(n – 2) = 1 2 , …
1
2
3
1 · 2 · 3
1 · 2
n
n
we can expand (a + b) by using the 1 2 notation,
r
n
n
n
n
n
3
n
n
n
b +
i.e. (a + b) = a + 1 2 a n – 1 b + 1 2 a n – 2 2 1 2 a n – 3 b + … + 1 2 a n – r b + … + 1 n –1 2 ab n – 1 + b ,
r
r
1
2
3
or in short,
n n
n
(a + b) = ∑ 1 2 a n – r r +
b , n  Z .
r = 0 r
The above result is called the binomial theorem.
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02 STPM Math T T1.indd 124 3/28/18 4:21 PM