Video Transcript
Find 𝑐, given the arithmetic sequence 𝑎, 37, 𝑏, 61, 𝑐.
Given the arithmetic sequence 𝑎, 37, 𝑏, 61, 𝑐, let’s think about some properties of an arithmetic sequence. In an arithmetic sequence, the values between 𝑎 sub one and 𝑎 sub 𝑛, the values between the beginning and ending terms, are called arithmetic means. The second term is the arithmetic mean of the first and third terms. And that means 𝑎 sub one plus 𝑎 sub three divided by two will equal 𝑎 sub two. The second term is halfway between the first and the third term. And this means that 𝑏 is halfway between 37 and 61. 𝑏 is the arithmetic mean of 37 and 61.
And 37 plus 61 divided by two must be equal to 𝑏, which makes 𝑏 49. If we know the third term is 49 and the fourth term is 61, we also know that the fourth term is the arithmetic mean between the third and the fifth term. We’re saying 61 is halfway between 𝑏 and 𝑐, and that means 𝑏 plus 𝑐 divided by two must be equal to 61. If we set up this equation and plug in 49 for 𝑏, then multiplied both sides of the equation by two, we get that 49 plus 𝑐 equals 122. And by subtracting 49 from both sides, we see that 𝑐 is equal to 73.
The first method we’ve considered to solve this problem is using arithmetic means. We could also solve by finding the common difference. If we take the five terms in the sequence that we were given, because we know that they form an arithmetic sequence, we know that they have a common difference. That is, the difference between any two consecutive terms will be the same amount. To get from 𝑎 to 37, we add the common difference of 𝑑. But we can also say to get from 37 to 61, we need to add the common difference twice. That means we can write an equation that says 37 plus two 𝑑 equals 61. This will allow us to solve for the common difference.
To do that, we subtract 37 from both sides, and we find that two times the common difference equals 24. And then when we divide both sides by two, we find that the common difference is 12. We wanted to know what 𝑐 was equal to. 𝑐 will be equal to 61 plus the common difference. If 61 plus 𝑑 equals 𝑐 and we know the common difference is 12, then again 𝑐 will be equal to 73. Both methods are equally valid. They just rely on different properties of arithmetic sequences for solving. In the end, they both prove that, for this sequence, 𝑐 must be equal to 73.
