Question

Find the four arithmetic means between 12 and -13.


6, -1, -3, -8

6, 2, -3, -8

7, -1, -3, -8

7, 2, -3, -8

Answer 1

Arithmetic means are spaced by a constant difference. If the constant difference is \(a\), then in your case \( 12 - a - a - a - a - a = -13\). You have one more \(a\) than mean because you need to go beyond the final mean to the endpoint.

Can you solve that for the value of \(a\) and then determine the 4 arithmetic means? The first one will be \(12-a\), the second one \(12-2a\), third \(12-3a\), fourth \(12-4a\) and as a check, \(12-5a\) should equal \(-13\).

Answer 2

what?

Answer 3

If you have a series of numbers in an arithmetic series, each number is bigger or smaller by the same difference. One such sequence might be
1,3,5,7,9,11
Each term is 2 greater than the previous term. The 2 is called the constant difference.

So, if your problem was to insert 4 arithmetic means between 1 and 11, the answer would be 3,5,7,9 because they are the 4 numbers in that sequence between 1 and 11.

Your problem is to find 4 arithmetic means between 12 and -13. Hopefully it is clear that in this case you will be subtracting a constant number instead of adding. You need to figure out what number can be subtracted 4+1= 5 times to get you from 12 to -13. Having figured out what that number is (a in my discussion above), you would find 12-a for the first number in the sequence, 12-2a for the second number in the sequence, 12-3a for the third number in the sequence, and 12-4a for the fourth number in the sequence.

Any clearer?

Answer 4

not really but thanks anyway.

Answer 5

Well, let's try a different approach. One way you can see if a sequence is an arithmetic sequence (which arithmetic means are, by definition) is you subtract one number from the next. If the sequence is arithmetic, you always get the same result.

1,3,6,9 is not arithmetic because
9-6=3
6-3=3
3-1=2
not all the same difference
0,3,6,9 is arithmetic because
9-6=3
6-3=3
3-0=3
all the same difference.

So, given that this is a multiple-guess question, and you have answers to look at, you could test them out to see which one works. The correct one will have the same difference between each number in the answer, and if you add that difference to the biggest number, you'll get 12, and if you take it away from the smallest number, you'll get -13.

Answer 6

Here's yet another way to think of it. Plot 12 and -13 on a number line. Now, evenly space 4 more points between them. Where are those points?