Sequences and Series

Sequences

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:

Solution:

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:

Sequence Examples

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

Solution:

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

and

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:

Solution:

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:

Series

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:

for example

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 a_n
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