The first term of an arithmetic
progression is five, the last term is 45, and the sum of all its terms is 400. Find the number of terms and the
common difference of the arithmetic progression.
To begin this question, we can
first write down the key information that we’ve been given. The question tells us that the
first term of the arithmetic progression is five. We’ll call this term 𝑎 one, with
the one representing the number of the term. The question also tells us that the
last term is equal to 45. We’ll call this term 𝑎 𝑛. Again, here the 𝑛 represents the
number of the term, and currently we don’t know what this is. We’re also told that the sum of all
the terms is 400, and we’ll call this 𝑆 𝑛.
Now it’s really important to also
note that we’re dealing with an arithmetic progression. What this tells us is that
successive terms will grow or shrink by the same value each time. In other words, the difference
between successive terms of our arithmetic progression will always be the same. We’ll come back to the implications
of this later on.
To begin, we’re gonna focus on
finding the number of terms in our arithmetic progression. In our notation, we have defined
that our progression has a total of 𝑛 terms, with 𝑎 𝑛 being the last term. And we know that the sum of all of
these 𝑛 terms is 400.
Now you may be familiar with the
following formula, which says the sum of all 𝑛 terms in an arithmetic progression
is equal to the number of terms, 𝑛, divided by two multiplied by the first term add
the last term. Now for this video, we’re not gonna
go into the full proof of this formula. However, you might be able to gain
an intuitive understanding with a small rearrangement. Looking at this rearrangement, we
can see that we have taken an average of our first term and our last term by adding
them together and dividing by two.
Now earlier on, we mentioned that,
for an arithmetic progression, the difference between successive terms is the
same. Since the terms change by the same
value each time, finding the average of the first and the last terms is essentially
the same as finding the average of all the terms in the progression. We then multiply this average by
the total number of terms, 𝑛. Thinking about the formula in this
way, we might be able to see that multiplying the average value of each of our terms
by the total number of terms will indeed give us a sum of all the terms.
Now that we understand this, let us
now use this formula to find 𝑛, the number of terms. We first substitute in the values
that we already know: 400 as the sum of all the terms, five as our first term, and
45 as our last term. To get rid of this factor of a half
on the right-hand side of our equation, we multiply both sides by two. And we can also simplify our
brackets, since five add 45 is equal to 50.
We now have that 50 times 𝑛 is
equal to 800. And we therefore divide both sides
of our equation by 50, to see that 800 divided by 50 is equal to 𝑛. 800 divided by 50 is 16, and we’ve
therefore found that the total number of terms 𝑛 is 16.
Now that we found this, we’re able
to tidy things up by replacing the 𝑛 with 16. 𝑎 16, or the 16th term of our
arithmetic progression, is 45. And the sum of all 16 terms is
equal to 400.
Now let’s move on to finding the
common difference between our terms, which we’ll call 𝑑. Now we already know the value of
our first term, 𝑎 one, is five, but what about the value of our second term, 𝑎
two? Well, since we’ve defined our
common difference to be 𝑑, the value of 𝑎 two will be the value of 𝑎 one plus the
common difference 𝑑.
We already know the value of 𝑎 one
is five, so 𝑎 two is five plus 𝑑. By similar reasoning, the value of
our third term, 𝑎 three, will be the value of the previous term, 𝑎 two, plus 𝑑
again. We can use our previous line of
working to substitute in for 𝑎 two, which is 𝑎 one plus 𝑑. And we know that 𝑎 one is five, so
we have that 𝑎 three is five plus two 𝑑.
Here we can begin to spot a
pattern. And by following this pattern, we
can build up a formula for any number term in the arithmetic progression. Here we see that, for our second
term, 𝑎 two, we have the first term, 𝑎 one, plus one lot of the common
difference. And for our third term, 𝑎 three,
we have our first time, 𝑎 one, add two lots of the common difference. Using this pattern, we’re able to
say that the term numbered 𝑘 in our arithmetic progression will have the value of
our first term plus 𝑘 minus one times the common difference 𝑑.
Using this rule that we found, we
should now be able to calculate the common difference, 𝑑, since we know the value
of the 16th and final term of our arithmetic progression. We first substitute 𝑘 equals 16
into our formula. We know that 𝑎 16 is equal to 45
and that 𝑎 one is equal to five. 16 minus one is 15, so we now have
that 45 equals five plus 15𝑑. We subtract five from both sides,
to find that 40 is equal to 15𝑑, and then divide both sides by 15, to find that 40
over 15 is equal to 𝑑.
Finally, we can simplify our
fraction by dividing the top and bottom half by five. And we find that the common
difference, 𝑑, is equal to eight over three. We have now fully answered the
question. And we found that the number of
terms, 𝑛, is 16 and the common difference, 𝑑, is eight over three for the