Core Maths C1 Revision Notes - Mr Barton Maths

4 Sequences and series
A sequence is any list of numbers.
Definition by a formula x n = f(n)
Example: The definition x n = 3n 2 – 5 gives
x 1 = 3 × 1 2 – 5 = –2, x 2 = 3 × 2 2 – 5 = 7, x 3 = 3 × 3 2 – 5 = 22, … .
Definitions of the form x n+1 = f(x n )
These have two parts:– (i) a starting value (or values)
(ii) a method of obtaining each term from the one(s) before.
Examples: (i) The definition x 1 = 3 and x n = 3x n –1 + 2
defines the sequence 3, 11, 35, 107, . . .
(ii) The definition x 1 = 1, x 2 = 1, x n = x n –1 + x n –2
defines the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 65, . . . .
This is the Fibonacci sequence.
Series and Σ notation
A series is the sum of the first so many terms of a sequence.
For a sequence whose nth term is x n = 2n +3 the sum of the first n terms is a series
S n = x 1 + x 2 + x 3 + x 4 . . . + x n = 5 + 7 + 9 + 11 + . . . + (2n + 3)
n
x i
i=
1
This is written in Σ notation as S n = ∑
n
= ∑ ( 2i
+ 3)
and is a finite series of n terms.
i=
1
An infinite series has an infinite number of terms S ∞ = ∑ ∞
i=1
Arithmetic series
An arithmetic series is a series in which each term is a constant amount bigger (or smaller)
than the previous term: this constant amount is called the common difference.
Examples: 3, 7, 11, 15, 19, 23, . . . . with common difference 4
28, 25, 22, 19, 16, 13, . . . with common difference –3.
x
i
.
Generally an arithmetic series can be written as
S n = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + . . . upto n terms,
where the first term is a and the common difference is d.
The nth term x n = a + (n – 1)d
The sum of the first n terms of the above arithmetic series is
12
C1 14/04/2013 SDB