Arithmetic sequences and series
To best understand sequences and series, we will start by explaining what they are.
- A sequence is a list on numbers in a specific order and pattern. For example; 1, 4, 9, 16, 25, … The numbers are in order and there is a pattern here that can be written as n2 starting from 1.
- A series is a sum of a given sequence. For example: S5 = 1 + 4 + 9 + 16 + 25 from the sequence in 1. The plus sign in between all the terms (elements of the sequence) denotes a series.
Arithmetic sequence and series
Derivation of formulae
An arithmetic sequence has a common difference between any two consecutive terms. Look the following example:
The difference between two consecutive terms is 2 for this sequence. This means that if I want to increase the sequence by one term I must add a 2. If I must find the previous term I must subtract by 2. Let us write this using a mathematical expression.
Before you continue, make sure that you understand how we arrived to the general sequence above. Further derivations will assume that you understand this. a is the first term and d is the common difference. Note that the addition of each element of the sequence is in relation to the first term.
The question we must ask then is how can we find any term in the sequence. This we can do by generating a general equation to find any term on the arithmetic sequence. We will introduce a new symbol called “n“. n will represent a term in a sequence. The first term will have n = 1, the second term n = 2 and so on.
This is a very important equation. Make sure you are clear on how we arrived to this equation as from now on we will be using it a lot.. One more symbol to add is for the last term and then we are ready to do some examples. The nth term and Tn are the same thing.
Practise: Arithmetic Sequences
All practise questions in this course will be taken from previous papers either provincial or national in order to make sure that you are accustomed to the NSC standard.
Try to do the question first before you check our suggested answer, it’s a good work ethic 🙂 .
Given: T2 = 8 | T5 = 10
We must use the equation general term equation derived above to get the equations for each term and then solve for a and d.
Tn = a + (n – 1)d
8 = a + (2 – 1 )d
8 = a + d : a = d – 8 … equation 1
Tn = a + (n – 1)d
10 = a + (5 – 1 )d
10 = a + 4d … equation 2
We now solve a and d simultaneously. Substrate equation 1 from equation 2.
T5: a + d = 8
T5: a +4d= 10
therefore 3d = 2 (a’s cancel each other).
- Tn = a + (n – 1)d The nth term and Tn are the same thing.
2. Calculating T43
I believe that you now have a better understanding of arithmetic sequences. Now we are transitioning into arithmetic series. Again, make sure that you can
- derive the expression for the sequence,
- the general formulae for the nth term (Tn) and
- the equation for the last term ( l ).
Arithmetic series come into play when we want to add the terms of a sequence. But before we look at the practical examples of how they are used, let us begin by deriving the general equation for finding a sum of a sequence.
Remember that the general sequence is described as a, a+d, a + 2d, a + 3d + …
The sum (Sn) for n terms will then be:
All we did was to add the terms in order to get the series. Now remember that the last term is represented by the letter l. So we can write the series in terms of l. By this I mean that we will write the series from the last term ( l ) all the way to the first term. This is how you can do that:
DO NOT PASS THIS SECTION TOO QUICKLY. Make sure that you understand how we arrived at these two general equations. More explanation follows. The first sum equation we added all the terms of the sequence that you are familiar with already. On the second equation we became smart. Instead of doing the terms from first to last, we added the terms from the last one all the way to the first one. Understood? I hope you do. The next and final step is to add these two equations. Remember they are the same it is just that one is written from first to last while the other is written from last to first. Let’s add the up …
Remember that l is the last term with and equation l = a + (n-1)d, substitute it on the equation of Sn above. This is what we get:
Are you happy? Go through the explanation above again and again and make sure that you can derive the equation of the arithmetic series.
Practise: Arithmetic Series
Do you remember what I said about simultaneous equations? If you are still struggling with them, jump to the section on functions, get comfortable and then return back. TIP: To get the sum on number 2, you must first get a and d (or maybe l ) before you can get the sum.
- We must first calculate a and d.
- Now with the values of a and d calculated, we can then easily calculate the sum.