# Video: Finding Five Arithmetic Means between Two Numbers

Find 5 arithmetic means between 7 and 19.

02:18

### Video Transcript

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 difference 𝑥.

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.