I am currently in school and we are learning about Arithmetic Sequences. In one of the sections in our learning material, the arithmetic mean is defined as the term of the arithmetic sequence that is between two given terms. For me, at least, I know this is wrong. In all the reliable sources I found about the definition of arithmetic mean, its definition is the sum of the numbers divided by the count of the numbers. Here, an example is given that the arithmetic mean(s) of 2, 4, 6, 8, 10 are 4, 6, and 8 (which should be 6). Our material also mentions about inserting arithmetic means between two numbers like "Insert 4 arithmetic means between 7 and 47," wherein the answer would be 15, 23, 31, and 39 because it will form an arithmetic sequence. As far as I know I don't know anywhere where this definition is taught. I think this is stupid. So my questions are,
- Is this really accepted? or where is it accepted?
- What should I do about it? We have been taught this 2 months ago but no one bothered to ask the teacher about it that time
2 Answers
$\begingroup$I don't personally love this terminology, but there is a reason for it. In your arithmetic sequence $$2,4,6,8,10,$$ each term between $2$ and $10$ is the arithmetic mean of the two numbers adjacent to it, i.e.\begin{align} \frac{2 + 6}{2} &= 4\\ \frac{4+8}{2} &= 6\\ \frac{6+10}{2} &= 8, \end{align}and so in some sense it makes sense to call these the arithmetic means of the sequence between $2$ and $10$. In general, given an arithmetic sequence $$a_{1},a_{2},\dotsc,a_{n},\dotsc$$ with common difference $a_{2} - a_{1} = d$, the arithmetic mean of any two terms of the form $a_{k}$ and $a_{k+2}$ will be the middle term, $a_{k+1}$:$$\frac{a_{k} + a_{k+2}}{2} = \frac{a_{k} + a_{k} + 2d}{2} = \frac{2(a_{k} + d)}{2} = a_{k} + d = a_{k+1}.$$
$\endgroup$ 2 $\begingroup$Why should the $8$ (which is arithmetic mean of $6$ and $10$) be instead $6$ as you say in the parentheses of your example starting "Here, an example"? It looks like in that sequence of your cited example that each term except first and last is indeed the arithmetic mean of the terms before and after it. The inserted sequence doesn't seem to be related to any use of arithmetic means of more than two terms.
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