Sequences and Series
A sequence in mathematics is defined as an ordered list of elements (usually numbers) whose order defines some underlying property of the list. The order of the elements is very important and changing even one element would change the meaning of the entire sequence.
The elements in a sequence are separated by commas and the length of a sequence is usually denoted by the letter n. The length of a sequence is equal to the number of terms in the sequence. Each element or term of a sequence is defined by its position in the sequences for example
In the above, each term is denoted by a with a subscript number denoting its position in the sequence. The first term is and the last term is where n is the length of the sequence. Sometimes the first term is also denoted by for example
The next term in a sequence is obtained by studying the relationship between the preceding terms in order to obtain a relationship which we can use to obtain the missing terms.
For example; find the 2 next terms (denoted by x and y)in the sequence below:
The first step is to try and obtain the relationship between the first and second term;
the difference between these two numbers is 1, 25 = 24 + 1
There seems to be no other obvious relationship between these two numbers, so we move on to study the second and third numbers;
The difference here too is 1 and 26 = 25 + 1. Now you should begin to see a trend in the sequence, the relationship between the terms seems to be that:
the next term is obtained by adding one to the previous (current) term
Let's study one more pair to confirm that the above is true:
The difference between these two numbers is 1 as we had suspected and 27 = 26 + 1 which proves the relationship we defined above.
Therefore, we can obtain x and y as follows:
such that the sequence becomes:
The above is a very elementary example. The relationship between the terms of a sequence may not always be so easy to spot. Here's a more challenging example:
Find the x and y in the sequence below
Ok, this may not be so challenging if you have already spotted the special property linking the terms in the above sequence. If you have, then let's indulge ourselves and pretend that you haven't. How would you obtain x and y?
The first step is to study the first two terms and try to see what kind of relationship exist between them;
if we were to subtract the first term from the second term 4 - 1 = 3. For now let's leave this pair and try to find out if the same relationship exists between the second and third terms
Taking their difference: 9 - 4 = 5
but 5 is not equal to 3. However 5 is the next odd number after 3 so let's assume that the relationship between two consecutive terms is that their difference gives the next odd number. This is true for the first two pairs we've seen as the next odd number after 3 is 5. If the relationship is to hold for the whole sequence, the difference between the terms in the next pair should be equal to the next odd number, which is 7.
So now let's look at the next pair to check if the above relationship does indeed hold true:
taking their difference: 16 - 9 = 7
The relationship is true so far but to be on the safe side, let's look at the last pair
taking their difference: 25 - 16 = 9
9 is the next odd number after 7. So now we have confirmed that the relationship between two consecutive terms is as follows:
the next term is obtained by adding the next odd number to the current term
From the above we can see that
Such that the sequence becomes
We could also have obtained x and y by observing that all the terms in the sequence are squares of the numbers from 1 to 5, so it would appear that x would be the square of 6 which is 36 and y would be the square of 7 which is 49.
A way we can find this relationship is to plot the points on a table in terms of x and y. The x value would be the term and the y value would be the value for that term, just like a function! We can see that the relationship between the y values is obtained by adding the next odd number. Going a step further, we can see that the relationship between each odd number goes up by 2, which is always consistent. This means that the relationship of our function is quadratic! We now know the function is f(x) = x2.
The relationship between two consecutive terms is not always a difference relationship, it might be multiplication or division or something more complicated as we'll see when we get to series.
Here is a rather challenging example:
Find the next 2 terms (x and y) in the sequence below:
By inspecting the above, it should be easy to see that subtracting one term from the other is not going to produce some clean straight forward relationship. The differences between each pair of consecutive terms are:
There is clearly no definable relationship between those numbers above. So straight away we dismiss using 'difference' as the way to come up with the relationship in the sequence in question.
The next options to try are multiplication and division. If we divide one term by the previous term, will there be some relationship in the numbers that we obtain?
If we divide the terms in consecutive pairs we end up with
which leads to:
looking at only 1,2,3,4,5 the relationship should be very clear. The numbers are increasing in the increment of 1, so after 5 we should have 6 then 7. So we can say the the next term is obtained by multiplying the current term by the position of the next term.
Therefore since x is the 6th term, we can obtain the value of x by multiplying 120 by 6
and y is obtained by multiplying x with 7
The sequence thus becomes:
A Series in mathematics is defined as the sum of the elements in a sequence. A series is obtained by adding up all the terms in a sequence as shown in the example below
A series is denoted by the letter S and the length is also denoted by the letter n as in:
Since a series is a sum, we can also represent it mathematically as
where i represents the position of the term a. The above means that a series is the sum of all terms from a1 to an where n is the length of the sequence.
Since a series is a sum of a sequence, the terms in a series also have a special relationship that defines some underlying property of the series and can be used to find the next term in the series is the sum of a given number of terms in the series. Most series that we deal with are of a finite nature, they don't go on forever like some sequences do. We'll soon see why it is important to have series that terminate.
In the section on Sequences above, we established the relationship between consecutive terms of a sequence and we have already said that the same applies to the terms of a series. So knowing this, we can find any number of terms in a series and find its sum.
Quiz on Sequences and Series
This sequence is known as the Fibonacci sequence. Its a very popular sequence in math. Consecutive terms are obtained by summing up the two previous terms. The first two terms are given as zero and one ie
The next term required is thus obtained from
So the sequence becomes
This is called the look and say sequence, the mathematical formula that relates the consecutive terms is quite complicated but the sequence itself is simple.
Since the first term is one(1) the next term is obtained by saying what the previous term was i.e 1, one 1, two 1s, one 2 one 1, so it follows that the next term should be one 1 one 2 two 1s. i.e.
To find the sum of the series all we do is add up all the terms