Question Video: Finding the Number of Arithmetic Means Inserted between Two Numbers under a Certain Condition Mathematics

Video Transcript

Find the number of arithmetic means inserted between two and 254 given the ratio between the sum of the first two means and the sum of the last two means is 11 over 245.

Let’s think about what we know. We have some sequence where the first term is two and the last term is 254. In sequences like this, the second term is equal to the first mean and the third term is equal to the second mean. We’ll let 𝑎 be our first mean and 𝑏 be our second mean. We know to get from our first term to our second term, there must be a common difference of 𝑑. The same thing is true. To get from our second term to our third term, we need to add a common difference of 𝑑. But how should we label our last two means?

If we began at our final term 254, the last mean will be negative 𝑑 away from the last term. We can let 𝑒 be our last mean. And if we take the last mean and subtract the common difference of 𝑑, we get a second to last mean. We’ll write our first two means in relation to the first term. 𝑎 will be equal to two plus 𝑑. And 𝑏 would be equal to two plus two 𝑑. And then we can write 𝑒 and 𝑓 in terms of our last value 254. 𝑒 will be equal to 254 minus 𝑑. And 𝑓, the second to last term, would be equal to 254 minus two 𝑑.

Now our ratio is the sum of the first two over the sum of the second two. And that means we want to add 𝑎 and 𝑏 and add 𝑒 and 𝑓. For the first two means, they sum to four plus three 𝑑. And for the last two means, they sum to 508 minus three 𝑑. We’ll take these two expressions and set them equal to our ratio of 11 over 245. We have the sum of the first two means, four plus three 𝑑, over the sum of the last two means, 508 minus three 𝑑, which must be equal to 11 over 245.

When we cross multiply, we get 245 times four plus three 𝑑 is equal to 11 times 508 minus three 𝑑. So we distribute, which gives us 980 plus 735𝑑 on the left and 5588 minus 33𝑑 on the right. Next, we add 33𝑑 to both sides of the equation. And then we need to subtract 980 from both sides to get 768𝑑 is equal to 4608. When we divide both sides of the equation by 768, we find that 𝑑 equals six. We now know the common difference is six. And we’ll need to use this to figure out how many means are between two and 254. That means we’ll need to figure out how do we get from two to 254 in increments of six.

Algebraically, we can write that as two plus 𝑥 times six is equal to 254 and then solve for 𝑥. When we do that, we find that 𝑥 equals 42. That means that we are taking 42𝑑 and adding it to our first term of two to get to 254. But to get to the last mean, we only need to add 41𝑑. Because two plus 41𝑑 equals the last mean, there are 41 means between two and 254.