### Video Transcript

Describing Relationships and
Extending Terms in Arithmetic Sequences. Here is an example of an arithmetic
sequence. A sequence is any ordered list of
numbers. An arithmetic sequence is a
sequence found by adding the same number to the previous term. For example, you add nine to
28. And that equals 37. You add nine to 37. And that equals 46. Each of these numbers are terms in
the sequence.

Someone might ask you, find the
next term in this arithmetic sequence.

When they say, find the next term,
we know that they’re looking for the value that comes after 64 here. And because we’re also given the
information that it’s an arithmetic sequence, we also know that we will be adding to
find the next value. Each step in this sequence has been
adding nine to the previous term. This lets us know that whatever 64
plus nine is, that’s our next term. The next term in this sequence is
73.

Here’s an example of a simple
arithmetic sequence. We’re adding two each time.

When we’re dealing with sequences,
each of the terms also has a position. You can see the labels here. The two is in first position. The four is in second position. Six is in third position and then
fourth and fifth, and so on. Positions are really important. Let’s take a closer look at what is
happening here with these positions. Position one has a value of
two. Position two has a value of
four. To get from position one to
position two, we need to add two. And to go from position two to
position three, we need to add two. There is a relationship here
between the position of a term and the term’s value. This column shows us that, to solve
for position one, we multiply one times two. To solve for position two, the
value, we take two. And we multiply it by two. To solve for position three, you
multiply three by two to get six.

In the operation, the first number
we’re looking at is the position number. The two is the number that we’re
adding to each term. Some of you might be wondering,
Well why would I take the time to figure out the multiplication when I could just
add two to six? Six plus two is eight. That’s pretty simple. And that’s true. If I wanted to find the next number
in this sequence, I would probably just say 10 plus two is twelve. The sixth position is twelve. But if the question said something
like what is the 80th position in this sequence, you don’t wanna add two 80
times. Now we’re going to add the number
80, the position 80, to our table. We would follow the same
operation. In the 80th position of this
sequence is the number 160. Finding that by multiplying 80
times two is significantly faster than trying to add two each time 80 times.

You might also come across the
question: what is an algebraic expression for finding terms in this sequence? In that case, we don’t know what
position we’re looking for. We’re looking to write an
expression that can be used for all positions. When dealing with positions and
sequences, we usually use the letter 𝑛 to represent an unknown position. We could find the value of a term
in position 𝑛 by multiplying the position by two, 𝑛 times two. We could say 𝑛 times two for our
expression, or simply two 𝑛. We can plug in any position number
here and find the term that would be in that position. We could plug in position 100,
position seven, position 15. It doesn’t matter. This expression works for solving
this sequence. Here’s another example.

Find the next term in this
sequence. 51, 102, 153. What’s next?

The first question we should ask is
what is being added to each term. Last time, that was really easy
because it was two. If you don’t immediately recognise
what’s being added, here’s what you can do. You can take the term in the second
position and subtract the term in the first position. 102 minus 51 is 51. You could also subtract 102 from 53
or the position two number from the position three number. Both of these equal 51. 51 is what is being added each
time.

To find the next term, we need to
take the third term, 153, and add 51 to that. The next term in this sequence is
204. Here’s what a table for this
sequence would look like. The operation here would be to take
the position and multiply it by 51 because 51 is what we’re adding here. 51𝑛 would be the expression that
we could use to take any position and find its value.

Here is our last example.

Find the 70th term in the following
sequence. Six, 12, 18, 24, ….

We’re trying to answer the question
what is being added to each term. And we know that here each term is
six more than the previous term. But I’m not just trying to find out
what are we adding to each term. I need to find out what expression
can I use to find the 70th term. I notice the pattern is that you
take the position. And you multiply it by six. So in order for me to find the 70th
term, I need to multiply that by six. When I do that, I have a solution
of 420. I also know that I can find any
term in this sequence by taking the position 𝑛 and multiplying it by six.

That expression is important. That’s all for this video. Now you can go try some sequences
on your own.