Find five arithmetic means between
seven and 19.
We have two values, seven and
19. And we’ve been told that there are
five arithmetic means between them, which we could mark 𝑎, 𝑏, 𝑐, 𝑑, 𝑒. Our first mean is 𝑎. And we know that if 𝑎 is the first
mean, the distance from seven to 𝑎 must be equal to the distance from 𝑎 to 𝑏. And so we can say if seven plus 𝑥
equals 𝑎, then 𝑎 plus 𝑥 must equal 𝑏. Moving on, 𝑏 is the second mean,
which means that the distance from 𝑎 to 𝑏 must be equal to the distance from 𝑏 to
𝑐. If 𝑎 plus 𝑥 equals 𝑏, then 𝑏
plus 𝑥 must equal 𝑐. This must be true for all five of
our arithmetic means between seven and 19. They must have a common
difference. And we’re calling that common
If there’s a common difference of
𝑥 between all five of these arithmetic means, there will also be a common
difference of 𝑥 between the last mean 𝑒 and the end of our sequence 19. We can use this information to set
up an equation. We can create a relationship
between seven and 19 using the common differences. To get from seven to 19, if there
are five arithmetic means between them, you need to add six 𝑥. So we’re saying seven plus six 𝑥
must be equal to 19. This equation will allow us to
solve for that common difference.
We subtract seven from both sides
of our equation and find six 𝑥 equals 12, which means 𝑥 equals two. And if 𝑥 equals two, 𝑎 equals
nine, 𝑏 equals 11, 𝑐 equals 13, 𝑑 equals 15, and 𝑒 equals 17. And the five arithmetic means
between seven and 19 are nine, 11, 13, 15, and 17.