Quick answer: To insert k arithmetic means between a and b, divide b − a by k + 1. There are k inserted terms but k + 1 equal gaps between the endpoints.
Worked example
Insert three arithmetic means between 4 and 20.
- There will be five terms in total, with four gaps.
- d = (20 − 4)/4 = 4.
- Add d repeatedly: 4, 8, 12, 16, 20.
- The inserted means are 8, 12 and 16.
Check the result
All four consecutive differences equal 4, and the final endpoint is 20.
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Common mistake to avoid
Dividing by 3 here would count inserted numbers instead of gaps. A negative difference is valid when the endpoints decrease.
Try it yourself
1. Insert two means between 5 and 14. 2. Insert one mean between −6 and 8.
Answers and reasoning
1. d = 9/3 = 3; means 8 and 11. 2. d = 14/2 = 7; the mean is 1, also (−6 + 8)/2.
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