Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series

P1^

3

The expansion of (1 + x)n

When deriving the result for n

r









you found the binomial coefficients in the

form

1 n nn(– ) nn nnnn n

!

(– )( –)

!

(– )( –)(– )

!

1

2

12

3

12 3

4



This form is commonly used in the expansion of expressions of the type (1 + x)n.

() 11 (– 121 )( –) 1212 (– 3 )( –

2 3

+=xnn ++x nn× xn+ nn×× xn+ n 1112234 3

)(nn–)(– )x^4

×× × +...

+

×

nn(– 1 )xnnn––++xxn

12

(^211)
ExamPlE 3.16 Use the binomial expansion to write down the first four terms, in ascending
powers of x, of (1 + x)^9.
SOlUTION
() 11 +=xx^9 ++ (^99812) ×× x^2 + 12987 ×××× 3 x^3 +...
= 1 + 9 x + 36 x^2 + 84 x^3 + ...
The expression 1 + 9 x + 36 x^2 + 84 x^3 ... is said to be in ascending powers of x,
because the powers of x are increasing from one term to the next.
An expression like x^9 + 9 x^8 + 36 x^7 + 84 x^6 ... is in descending powers of x, because
the powers of x are decreasing from one term to the next.
ExamPlE 3.17 Use the binomial expansion to write down the first four terms, in ascending
powers of x, of (1 − 3 x)^7. Simplify the terms.
SOlUTION
Think of (1 − 3 x)^7 as (1 + (− 3 x))^7. Keep the brackets while you write out the terms.
(( 13 +=–)x)(^717 ++–) 3 x^7612 ×× (– 3 x)(^2 +^7612 ×××× 35 – 3 x))^3 +...
= 1 – 21x + 189 x^2 – 945x^3 + ...
The power of x is
the same as the
largest number
underneath.
Two numbers on top,
two underneath. Three numbers on top,
three underneath.
Note how the signs
alternate.