### Video Transcript

In any sequence pattern, if the
difference between any two successive terms is a fixed number π, then this is
an arithmetic sequence. Consider the sequence one,
four, seven, 10, and so on and then answer the following questions. Is the sequence arithmetic? What is the value of π? What is the general term of
this sequence with π is greater than or equal to zero?

To start this question, weβre
given a little reminder about the definition of an arithmetic sequence. And itβs one which has a
difference or common difference between any two successive terms the same. This word successive is like
the word consecutive. It simply means two terms where
one immediately follows the other one. We are asked to consider the
sequence one, four, seven, 10, and so on. If we look at the first
question, weβre asked if the sequence is arithmetic. So if itβs arithmetic, it will
have a common difference between any two consecutive terms.

So if we wanted to find the
difference between the first and the second term, we would work out four take
away one, which of course is three. To find the next common
difference between the third term and the second term, we would work out seven
take away four, and thatβs also three. In the same way, the difference
between 10 and seven is also three. As we have a common difference,
then we do have an arithmetic sequence. And we can say yes as the
answer for the first part of this question. The second part of the question
asks us, what is the value of π? π is the common difference, and weβre
reminded of that in the question text. And itβs also nice and easy to
work out; itβs three. And thatβs the second part of
the question answered.

In the final part of this
question, weβre asked to find the general term of this sequence with π greater
than or equal to zero. Remember that the general term
is often seen as the πth term. The fact that π is greater
than or equal to zero means that the index of this sequence should start with π
equals zero. So actually, this sequence
begins with a zeroth term. Then weβd have the first term,
then the second term and the third term and so on. We should recall that there is
a formula to help us find the πth term of an arithmetic sequence. The πth term π sub π is
equal to π plus π minus one multiplied by π, where π is the first term and
π is the common difference.

When we look at the sequence
one, four, seven, 10, and so on, the first term is really the same as π sub
one. So the value of π, which we
plug in, will be four. And as the value of π, which
we worked out earlier, is three, then we add on π minus one multiplied by
three. When we distribute the three
across the parentheses, we get three multiplied by π, which is three π, and
three multiplied by negative one, which is negative three. Finally, when we simplify this,
we get four minus three, which is one, plus three π or three π plus one. And thatβs the answer for the
third part of this question: the general term or πth term of this sequence is
three π plus one.

But it is important to note
that this was because the index π was greater than or equal to zero. If the index had started with
one, i.e., π is greater than or equal to one, then we would have worked out the
πth term to be three π minus two. So itβs really important to
read the question to see if thereβs an indication that the index, in fact,
begins with zero. In this case, however, we can
give the general term as three π plus one.