When I started looking at “Sequences and Proof”, I fell into the trap of looking at some of the most difficult questions (A and A*) first. What a surprise to find that I couldn’t do them! I now think it’s especially important with this topic, to master the basic, simple stuff and then move onto more challenging areas such as proof using algebra.
So, the basics:-
A sequence is a list of numbers in a given order
Consecutive numbers are numbers next to each other
We call the numbers in a sequence, terms
This leads us to the most basic sequence:-
A liner sequence increases or decreases in equal sized steps.
For example:- 2, 4, 6, 8
In this sequence the terms increase by 2 in each step.
We can say: The nth term = 2n (i.e the 1st term = 1 x 2 = 2, the 2nd term = 2 x 2 = 4, the 3rd term = 3 x 2 = 6 and the 4th term = 4 x 2 = 8).
When you’re trying to understand a sequence it’s very useful to write the differences between each term under the sequence, like this:-
When you have a linear sequence if you look at differences between the terms to start to define the sequence, if the differences are all 2 then the sequence definition will include 2n, if the differences are all 3 then the sequence definition will include the term 3n, if the differences are all 4 then the the sequence definition will include the term 4n etc., etc.
This is best explained with a couple of examples:-
Find the nth term of the sequence 4, 6, 8, 10,.…
First look for the difference between each term:-
You now know that the sequence includes 2n. Set the information in a table including 2n:-
This method might seem over the top for simple linear sequences, but it shows its worth with more complicated examples:-
Find the nth term of the sequence 1, 9, 17, 25
So start by finding the difference between each term and then complete the grid as shown above:-
Most questions ask you to find the nth term but sometimes this is turned on its head. For example what is the 10th term if the nth term = 8n — 7? Just feed the actual number into the statement:
10th term = (8 x 10) — 7 = 73
Alternatively you might be asked to find out if a particular number is a term in the sequence, for example is 155 a term in the above sequence? Again just feed the number into the sequence definition.
If 155 is in the sequence, then 8n — 7 = 155
and 8n = 162
n = 162/8 = 20.25
So “n” is not a whole number and 155 is not in the sequence.
Another example using the same sequence, is 177 a term in the sequence?
If 177 is in the sequence then 8n — 7 = 177
and 8n = 184
n = 184/8 = 23
So “n” is a whole number and 177 is in the sequence.
The nth term of a quadratic includes an “n²”. To find out if a sequence is quadratic you check the second differences. So what is a second difference? We used the first differences between terms when we looked at linear sequences above. The second difference is the difference between the differences. This is best understood by using an example. Let’s invent our own quadratic sequence, n² + 1. So looking at the first few terms and the differences and the second differences we have:-
You may get a question where the answer is a multiple or fraction of n². In these cases the second difference will not be 2, but it will be constant. For example, find the nth term of this sequence:- 0.25, 1, 2.25, 4, 6.25
Proofs and Proof using algebra to follow.