### Video Transcript

If the first and last terms of an
arithmetic progression are five and 45, respectively, and the sum of all its terms
is 400, find its common difference.

Remember, an arithmetic progression
is a sequence of numbers such that the difference between each of the consecutive
terms is constant. We have two formulae which we can
apply to arithmetic progressions. The first is the formula for the
𝑛th term. It’s given by 𝑎 plus 𝑛 minus one
multiplied by 𝑑, where 𝑎 is the first term and 𝑑 is the common difference. The second is the formula for the
sum of the first 𝑛 terms in the sequence. It’s 𝑛 over two multiplied by two
𝑎 plus 𝑛 minus one 𝑑.

For a finite sequence such as ours,
this is sometimes written as 𝑛 over two multiplied by 𝑎 plus 𝑙, where 𝑙 is the
last term in the sequence. We can use this second version to
help us calculate the number of terms in our sequence. Since the first term is five and
the last is 45, we can substitute five and 45 into this formula, in place of 𝑎 and
𝑙. We also know that the sum of these
terms is 400. So we can equate this expression to
400 to get 400 equals 𝑛 over two multiplied by five plus 45. Five plus 45 is 50. And we can solve this equation for
𝑛 by first dividing everything by 50. 40 divided by five is eight. So similarly, 400 divided by 50 is
also eight.

Since we want to calculate the
value of 𝑛, we can next multiply everything by two. And that gives us 𝑛 is equal to
16. We now know that there are 16 terms
in our sequence. Now that we know there are 16
terms, we can use this information to work out the common difference.

Let’s now refer back to the first
formula, the one that helps us find any term in the sequence. We know that the 16th term now is
45. And we still have the first term of
five. Substituting these into the formula
for the 𝑛th term gives us 45 is equal to five plus 16 minus one 𝑑. 16 minus one is 15. So we get 45 equals five plus
15𝑑. To solve for 𝑑, we’ll subtract
five from both sides. And that gives us 40 is equal to
15𝑑. Next, we’ll divide through by
15. 𝑑 is therefore equal to 40 over
15.

Finally, we’ll need to simplify
this fraction by dividing both the numerator and the denominator by their common
factor, which is five. 40 divided by five is eight. And 15 divided by five is
three. So we have a common difference of
eight-thirds.