Video Transcript
Find the number of arithmetic means
inserted between eight and 238 given the sum of the second and the sixth means is
96.
Let’s think about what we know. We’re given eight and 238. And we’re trying to figure out how
many arithmetic means are between these two values given the sum of the second and
the sixth means is 96. We don’t know anything about the
terms between eight and 238 apart from that, but we do know that every consecutive
term will have a common difference. Our second mean is two common
differences away from eight. The second mean is the third
term. So let’s let 𝑎 be equal to our
second mean.
If 𝑎 equals our second mean, it’s
going to be equal to eight plus two times the common difference 𝑑. To go from our second mean to our
sixth mean, we need to add that common difference four more times. So we’ll let the sixth mean be
equal to 𝑏. We can write 𝑏 in terms of our
first term and our common difference. 𝑏 would be equal to eight plus six
𝑑. If you start at eight and try to
get to 𝑏, you need to add the common difference six times. We know that the sum of the second
and sixth means is 96. 𝑎 plus 𝑏 must be equal to 96. We can plug in eight plus two 𝑑
for 𝑎 and eight plus six 𝑑 for 𝑏.
When we combine like terms, we find
that 16 plus eight 𝑑 equals 96. Subtracting 16 from both sides
gives us eight 𝑑 is equal to 80. And dividing both sides by eight,
we get that 𝑑 equals 10. This is not telling us how many
arithmetic means are between eight and 238. It’s only saying that the common
difference in this sequence is 10.
Now, we need to think of a way to
go from eight to 238 with a common difference of 10. We want to know if we start with
eight, how many sets of 10 do we need to add to eight to end up at 238? We subtract eight from both sides,
and we get 𝑥 times 10 equals 230. If we divide both sides by 10, we
get that 𝑥 is equal to 23. This means we’re saying eight plus
23 times the common difference equals 238. And that makes sense. The common difference is 10, so
eight plus 230 equals 238. But here’s where we have to be
really careful. This 23𝑑 gets us from the first
term to the last term, but our question wants to know the number of arithmetic means
between 238 and eight. And this means we need to go one to
the left of 238.
To get from 238 to the final mean,
we subtract 𝑑. If we let 𝑐 be equal to the final
mean between eight and 238, it’s located at eight plus 23𝑑 minus 𝑑. It would be equal to eight plus
22𝑑. And that 22 makes it the 22nd mean,
which means there are 22 means inserted between eight and 238.
