Video Transcript
If the sum of the second mean and
the fourth mean from an arithmetic sequence equals 16 and the seventh mean is more
than the third mean by eight, then the sequence is blank.
Let’s say we have some sequence
with the first term 𝑎. We could say the second term is 𝑏,
the third term is 𝑐, and continue on like this. We have to remember that if the
first term is 𝑎, the first mean is actually the second term in the sequence. If the second term is the first
mean, the third term is the second mean and the fifth term is the fourth mean. However, this string of variables
is not very helpful to us. It would be more helpful to write
these variables in terms of our first value in the sequence.
So let’s go back. If we let our first value in the
sequence be 𝑎, we know that our second term will be equal to our first term plus a
common difference. And this is a much better way to
write these values. Our second term would be equal to
𝑎 plus the common difference 𝑑. And our third term would be equal
to 𝑎 plus 𝑑 plus 𝑑. We can call that 𝑎 plus two
𝑑. Our fourth term would be equal to
𝑎 plus three 𝑑. And we need to think about the
values we are interested in. We have information about the
second mean, the fourth mean, the seventh mean, and the third mean.
We’ve already said that that second
mean would be equal to the third term and the fourth mean would be equal to the
fifth term. We can notice something interesting
here. The second mean has two units of
the common difference 𝑑 being added to the first term. And the fourth mean has four common
differences being added to the first term 𝑎. We see that with the third
mean. Three common differences are being
added. And we can use that to say that the
seventh mean would be the first term plus seven 𝑑.
Using these values, we can set up
some simultaneous equations to solve for the sequence. We know the second mean plus the
fourth mean equals 16. This means 𝑎 plus two 𝑑 plus 𝑎
plus four 𝑑 equals 16. We also know that the third mean
plus eight is equal to the seventh mean. So we write 𝑎 plus three 𝑑,
that’s the third mean, plus eight is equal to 𝑎 plus seven 𝑑, the seventh
mean. On the left, we can simplify to two
𝑎 plus six 𝑑 equals 16. And for our other equation, if we
subtract 𝑎 from both sides, we have 𝑎 minus 𝑎 on both sides.
We end up with three 𝑑 plus eight
equals seven 𝑑. By subtracting three 𝑑 from both
sides, we can see that eight is equal to four 𝑑. And by dividing both sides by four,
we find out that two equals 𝑑 or 𝑑 equals two. And then we wanna take that two for
our 𝑑-value and plug it into our second equation so that we have two 𝑎 plus six
times two equals 16. Two 𝑎 plus 12 equals 16. By subtracting 12 from both sides,
we get two 𝑎 is equal to four. And once we divide through by two,
we see that 𝑎 equals two.
Remember that 𝑎 represented the
first term in our sequence and our common difference here, our 𝑑-value, is two. This means that our second term
will be four, our third term will be six, and the pattern would continue. The arithmetic sequence described
here is the sequence with the first term of two and a common difference of two.
