Sequences
VIDEO COMING SOON…
In this post, we are going to be covering Sequences.
Before we get into the nitty-gritty, you need to understand what Sequences are and how they play a role within algebra.
Sequences are simply a pattern of numbers that follow a specific order or ‘sequence’. This pattern can be expressed using a formula with the unknown showing as ‘\(n\)’. ‘\(n\)’ represents the order position you are trying to find. For example. If we had a sequence that was \(2n+1\), if we wanted to know what number would be 3rd in the sequence, we would just sub in \(3\) as ‘\(n\)’ and the equation will give us what the 3rd number would be.
Finding the \(n\)th term
I’ve already explained what the nth term is in the introduction. It’s simply a way of determining a value for any position in a sequence by representing this value as ‘\(n\)’.
Linear Sequences
These sequences have a set difference between each number. This could be something like \(+3\) so our sequence would be something like \(1, 4, 7, 10,\) etc.
This is a linear sequence. The formulas for Linear Sequences will usually look something like ‘\(3n+1\)’ or ‘\(2n\)’.
In the example of a \(+3\) sequence, we would work out or nth term formula by putting the recurring difference next to ‘\(n\)’ – in our case this would be the \(+3\), so this would be ‘\(3n\)’. Then you compare the first number of the sequence, against the recurring difference. I.e. if we had the sequence of \(1, 4, 7, 10\), we can see that 1 is 2 less than 3 so our formula would be \(3n-2\).
We can prove this formula is right because we can check it – we know that the 4th number in the sequence is 10. However, we can sub in 4 as ‘\(n\)’ and see if this gives us 10 as well.
\(3(4)-2 = 12-2=10\) which is correct.
Quadratic Sequences – what are they?
This sounds a bit complicated but it’s just a little expansion on the Linear Sequences method.
So with the Linear Sequences, our difference is a set number such as \(+3\). However, for Quadratic Sequences, we need to note down the differences between each sequence value to give ourselves a new sequence, which we then will have a set difference for. It’s a little confusing to explain on paper so let’s look at an illustration to clear things up.
Look at the below Quadratic Sequence:
\(6, 10, 16, 24\) —> as you can see our difference is gradually getting exponentially greater. This is a key sign that we’re dealing with a quadratic sequence.
So let’s find our differences by subtracting each value from their consequent value:
\(10-6=4\)
\(16-10=6\)
\(24-16=8\)
Now if we put these differences into a sequence of their own i.e. \(4,6,8\), we can see that they’re in a linear sequence as they’re going up by 2 each time.
Quadratic Sequences – how do we show them as an equation?
With a linear sequence, we know that the difference gets put next to ‘n’ when expressing the sequence as a formula. However, as we’ve gone a layer deeper and found the differences of our differences we will be starting out the formula with an \(n^2\) rather than an \(n\).
Unlike linear sequences, you don’t just put our newly found difference of 2 next to the \(n^2\). You need to half it first. Therefore the start of our equation will be ‘\(n^2\)’.
Next, you need to take away the \(n^2\) term away from each corresponding value – I’ll demonstrate this below:
Original sequence term —> \(6, 10,16,24\)
nth term —> \(1,2,3,4\)
Therefore, \(n^2\) term —> \(1,4,9,16\)
So now we subtract the \(n^2\) term from the original sequence term, this gives us a new sequence term of —> \(5, 6, 7, 8\)
Using this new sequence term, we use the linear sequence equation method we spoke about earlier to find it’s equation.
As the first nth term is 1, we need to +4 to get to our first value in the new term, therefore our equation for this is \(n+4\).
Our final step is to put this all together —> \(n^2+n+4\).
We can quickly test by subbing in a value for \(n\) —> we know that our 3rd value in our original sequence is 16 so let’s sub in 3 as n and see if it gives us the value of 16:
\((3)^2+3+4 = 9+3+4 = 16\) —> happy days!
QUICK EXAM TIP
You may be shown a value and asked whether it will feature as a value in a given sequence. The key here is to make your sequential equation equal to the value shown.
If the value features in the sequence, you will return a value for n that is a whole number. However, if you are returnned a number that is not a whole number – it will not feature in the sequence.
Hopefully this all made sense. If you have any questions on this, please leave them in the comments and I’ll get back to you as soon as possible!
In the meantime, give these question worksheets a go and let me know how you get on.