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Q&A

What are the 2 arithmetic means of $x + y$ and $4x - 2y$?

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I'm currently learning arithmetic sequences, and I've gotten to the means. I'm answering an activity as a test to see if what I'm doing is right.

Here's an example through format:

First term = 10
Last term = 40
Arithmetic means = 5
Answer = {15, 20, 25, 30, 35}

I'm sure means work that way, even though I didn't research about it. Since I'm not that good with algebra, I'm having trouble handling this one:

First term =  x + y
Last term = 4x - 2y
Arithmetic means = 2

If not the full answer, can you at least give me a simple way to get it? The concept of $\sum$ still confuses me, so if you give me some answer using that, please explain it.

Revision

Alright, let's fix the question. What does an "arithmetic mean" mean? According to the book, it's the set of terms in order that are between the first and the last terms given which creates an arithmetic sequence.

If that's so, let's use the example with a correct answer:

First term = 10
Last term = 40
No. of arithmetic means = 5 (I should've pointed it out)
Answer = {15, 20, 25, 30, 35}
Arithmetic sequence = {10, 15, 20, 25, 30, 35, 40}
Common difference = 5

Let's try another with an example I thought of that's not in the book:

First term = 16
Last term = 72
No. of arithmetic means = 3
Answer = {30, 44, 58}
Arithmetic sequence = {16, 30, 44, 58, 72}
Common difference = 14

See what I mean?

Back to the original question, I can't figure out how to put $x$ and $y$ in this:

First term = x + y
Last term = 4x - 2y
No. of arithmetic means = 2
Answer = ?
(Current) arithmetic sequence = {x + y, ?, ?, 4x - 2y}
Common difference = ?

I might figure this out on my own; this has been hurting my head for a while now. I'll let you guys know.

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Terminology (5 comments)

2 answers

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I think I got it.

TL;DR: The answer is $2x$ and $3x-y$.

Apparently, since I was doing this at a state of "hard-thinking", I wasn't able to put much of this into mind simply because my brain was too exhausted with the problems, so I basically gave up when 2 letters on both sides showed up.

Now that I cleared my mind a bit more, I noticed something with both variables. $x$ was going up while $y$ was going down.

We can get the differences of each variable:

$$ x - 4x = -3x $$ $$ y - 2y = -3y $$

So now we know both variables have a difference of 3.

How was I able to get the arithmetic means of $10$ and $40$? By doing the same.

$$ 40 - 10 = 30 $$

So if we can get $10$ to $40$ using $30$, then it must be using $6$ ~~intervals (choose a better word for me) ~~ if the means were $5$.

$$ 30 / 6 = 5 $$

So all I had to do then was add $5$ to every mean starting from $10$, which immediately gave the sequence:

$$ 10, 15, 20, 25, 30, 35, 40 $$

So what if we did the same with $x + y$ and $4x - 2y$?

$$ 3 / 3 = 1 $$ $$ x, 2x, 3x, 4x $$ $$ y, 0, -y, -2y $$ $$ x + y, 2x, 3x - y, 4x - 2y $$

So there you have it. The two arithmetic means in this sequence were $2x$ and $3x - y$.

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This problems seems to be missing information, or alternatively it is simple.

Means here seems to be the difference between consecutive terms of an arithmetic sequence. Thus, the answer in the first exercise consists of the numbers 10+15, 10+25, 10+35, ..., 40-25, 40-1*5. That is, you start from the lower limit and continue in increments equal to the mean until you reach the upper limit.

Hence, the answer in the second is x+y+2, x+y+4, x+y+6, x+y+8, ..., all the way to 4x-2y-2.

However, this is not a very satisfactory answer, since the first term might be larger than the last term, and there is no guarantee that their difference, 3x-3y, is divisable by two; consider x = 1 and y = 0 for a simple case where things go wrong, and irrational numbers for even more fun.

I suspect the book is not very good or that it provides some context that is missing from the question here.

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