### Video Transcript

What value of 𝑘 will make 𝑘 plus
nine, two 𝑘 minus one, and two 𝑘 plus seven consecutive terms of an arithmetic
progression?

An arithmetic progression is a
sequence of numbers such that the difference between any two consecutive terms is
constant. Remember, consecutive simply means
one after the other. For example, one, three, five,
seven is an example of an arithmetic progression. It has a constant common difference
of two. And the first two consecutive terms
are one and three. This means that the difference
between the first two terms, that’s 𝑘 plus nine and two 𝑘 minus one, must be equal
to the difference between the second and the third term. That’s two 𝑘 minus one and two 𝑘
plus seven.

Remember, when we find the
difference, we subtract one from the other. So the difference between the first
two terms is two 𝑘 minus one minus 𝑘 plus nine. And the difference between the
second and third term is two 𝑘 plus seven minus two 𝑘 minus one. Let’s simplify each of these
expressions.

On the left-hand side, we get two
𝑘 minus one minus 𝑘 minus nine. It’s really important to be careful
when multiplying out the second bracket. Technically, what we’re doing is
multiplying everything inside this bracket by negative one, which is why we end up
with negative nine. A common mistake is to forget this
fact and simply write plus nine.

Similarly, on the right-hand side,
we’re multiplying the second bracket by negative one. That gives us negative two 𝑘 plus
one. When we simplify the left-hand side
of this expression, we get 𝑘 minus 10. Since two 𝑘 minus 𝑘 is 𝑘 and
negative one minus nine is negative 10. Two 𝑘 minus two 𝑘 is zero. So we end up with eight on the
right-hand side.

And we can solve this expression
for 𝑘 by adding 10 to both sides. Eight plus 10 is 18. And we can see that the value of 𝑘
that will make the terms consecutive terms of an arithmetic progression is 18.