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Sequences and Series
Sequences, Series and Sigma notation
Sequences
If you have a set of numbers T1, T2, T3,…where there is a rule for working out
the next numbers, we call this set a sequence.
Every number in the sequence is called a term and Tn is the nth term.
Examples:
a)3, 6, 9, 12… nth term = 3n
b)3, 9, 27, 81 … nth term =
c)1, 3, 6, 10… nth term = (the triangle numbers)
Series
A series is just when the terms of a sequence are added: T1 + T2 + T3 … Tn
For example: 1 + 2 + 4 + 8 + …
A finite series stops after a certain number of terms,
For example: 1 + 3 + 9 + 27 + 81, which has five terms.
An infinite series does not stop, (1/2)+(1/3)+(1/4)+(1/5)+(1/6)….
Sigma notation
We use the symbol sigma to define ‘the sum of the terms’ so that:
is the sum of all the terms where t takes the values between 2 and 6 inclusive.
Written out in full, this is:
The first term is when t = 3
i.e. 2 x 3 – 6 = 0.
The last tern is when t = 12
i.e. 2 x 12 – 6 = 18.
Arithmetic Progressions
Arithmetic Progressions
If you have the sequence 2, 8, 14, 20, 26, then each term is 6 more than the
previous term. This is an example of an arithmetic progression (AP) and
the constant value that defines the difference between any two consecutive
terms is called the common difference.
If an arithmetic difference has a first term a and a common difference of d,
then we can write
a, (a + d), (a + 2d), …{a + (n-1) d}
where the nth term = a + (n–1)d
Sum of Arithmetic series
The sum of an arithmetic series of n terms is found by making pairs each with
the value of the sum of the first and last term. (Try this with the sum of the first
10 integers, by making 5 pairs of 11).
This gives us the formula:
,where a = first term and l = last term.
As the last term is the nth term = a + (n – 1)d we can rewrite this as:
(Use the first formula if you know the first and last terms; use the second if you
know the first term and the common difference.)
Questions to try:
2.Every year the Queen presents special coins (Maundy Money) to a
number of selected people. The number of people receiving the coins in a
year is equal to twice the Queen's age in years.
Given that in 1952, the first year of the Queen's reign, her age was 26,
a) find an expression for the number of people receiving the coins in the
nth year of her reign,
b) calculate the total number of people receiving the coins from 1952 to
1998 inclusive.