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Year 5 Maths
Sequences and Series The Key Facts
A sequence is a set of numbers that displays an obvious pattern. By that we mean, it is easy to find
the next number in the sequence.
The terms in a sequence are often denoted by:
u1, u2, u3, ………….ur, …………………….. where ur is the rth term
A sequence can be represented in two ways:
a) The rule ur = 2r-1 allows us to write down the sequence for all r.
b) The iterative rule u1 = 2, ur+1 = ur + 2. The relationship between ur and ur+1 is called a
recurrence relation.
Convergence, Divergence and Periodic.
A sequence is convergent if it’s terms approach one finite value.
A sequence is divergent if it is not convergent.
A divergent sequence whose terms form a repeating pattern is called a periodic sequence.
When the terms of a sequence are added, a series is formed:
Eg 2 + 4 + 6 + 8 + 10 + …………. is a series.
A series can be defined more concisely by using the  symbol. Thus
means “the sum of all terms of the form r3, where r takes all integral values from 2 to 10”.
r 2
Arithmetic Progressions
A sequence where each term exceeds the previous term by a constant amount is called an Arithmetic
Progression (or A.P.) The constant amount is called the common difference d.
The nth term of an Arithmetic Progression is always given by:
a  n  1d (where a is the first term)
Therefore the fifth term will always be a + 4d, the seventeenth term a + 16d etc.
The sum of an Arithmetic Progression
The sum of an AP with n terms is given by:
Sn =
or S n 
na  l  where a is the first term and l is the last term
n2a  n  1d  where a is the first term and d is the common difference
Geometric Progressions
A sequence where each term is a common multiple of the previous term is called a Geometric
Progression (GP). The constant multiplying factor is called the common ratio.
The nth term of a Geometric Progression is given by:
ar n 1 where a is the first term and r is the common ratio.
The sum of a Geometric Progression
The sum of the first n terms of a GP is given by:
Sn 
a 1 rn
1 r
a is the first term, and r is the common ratio.