### Video Transcript

Which of the following expressions
can be used to calculate the sum of three consecutive odd numbers? Circle your answer. Three multiplied by two π plus
two, three multiplied by two π plus one, six π plus one, or two π plus nine.

So when weβre looking at the
question, we can see that we want to find the sum of three consecutive odd numbers
and then find one of our expressions that can be used to calculate this. Well, first of all, what does three
consecutive numbers mean? Well, consecutive numbers are
numbers that come directly after each other.

But in particular, we want
consecutive odd numbers. So that would mean, for instance,
one, three, five or five, seven, nine. So if we consider π to be a
number, how can we make sure that the number is going to be an odd number? Well, if we think about two π, we
know that two π is going to be an even number. Thatβs because two multiplied by
any number is even.

And two examples to show this out:
two multiplied by one, so thatβs an odd number, gives us an answer of two, which is
an even number. And two multiplied by four which is
an even number gives us an answer of eight which is again an even number.

So therefore, we can say that two
π minus one and two π plus one must be odd numbers because the numbers either side
of an even number are odd. As an example, we have one, two,
three.

So two is an even number. And then either side, so if we take
one away, we get one which is an odd number. And then, if we add one on, we get
three which again is an odd number.

Again, if we look at seven, eight,
nine, eight is an even number. Take one away, we get seven which
is an odd number. Add one on, we get nine which again
is an odd number.

So now, what we need to think about
is the sum of three consecutive odd numbers. So if we start with our two π
minus one, then the next odd number would be two π plus one, then the next odd
number would be two π plus three. And thatβs because weβre having a
difference of two each time.

And thatβs because between each odd
number, we would have a difference of two. So between one and three, there is
a difference of two. Between seven and nine, thereβs a
difference of two. And then starting from two π plus
one, the next two consecutive odd numbers would be two π plus three and two π plus
five.

Now the sum means adding them
together. So letβs add first of all two π
minus one, two π plus one, and two π plus three. So first of all, weβve got minus
one add one which is zero, then add three which is three. And then, we have two π add two π
which is four π add another two π which is six π.

So then, if we look at our next
three consecutive odd numbers, weβve got two π plus one add two π plus three add
two π plus five. So first of all, if we do one plus
three plus four plus five is nine. And again two π add two π is four
add another two π is six π.

Okay, so do either of these look
like the expressions that weβve got in the question? So as you can see that they donβt
fit any of the expressions that we have in our answer. Then, what we can do is
factorize.

Always watch out for factorization
in this type of question. And we know you can factorize cause
if we take a look at each of the answers to the sums, we can see that there are
common factors in the terms.

On the left-hand side, we can see
that a common factor of six π and three is three. And this is the highest common
factor. And then inside the bracket, weβre
gonna have two π plus one. And thatβs cause three multiplied
by two π is six π and three multiplied by one or positive one gives us positive
three.

And then again, if we look at the
right-hand answer to the sum, we can see that three would be the highest common
factor. So we take three outside the
brackets. And then inside the brackets, weβll
have two π plus three. And thatβs because three multiplied
by two π is six π and three multiplied by positive three gives us positive
nine.

And now, we can see if either of
these match our expressions. And yes! We can see that one of them does
because we got three multiplied by two π plus one which matches our second
expression.

And therefore, we can say that the
expression three multiplied by two π plus one can be used to calculate the sum of
three consecutive odd numbers.