Question Video: Finding the Number of Arithmetic Means Inserted between Two Numbers given the Sum of Two Means | Nagwa Question Video: Finding the Number of Arithmetic Means Inserted between Two Numbers given the Sum of Two Means | Nagwa

Question Video: Finding the Number of Arithmetic Means Inserted between Two Numbers given the Sum of Two Means Mathematics • Second Year of Secondary School

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Find the number of arithmetic means inserted between 8 and 238 given the sum of the second and the sixth means is 96.

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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.

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