Question 1185583: Insert the indicated number of harmonic means given the first and last terms. Show the solution.
1/2 and 1/6 [3] or insert 3
2/3 and 2/9 [2] or insert 2
5/2 and 5/27 [4] or insert 4
Answer by greenestamps(13200)   (Show Source):
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The terms of a harmonic sequence can be written as fractions with a constant numerator and denominators that form an arithmetic sequence.
Examples:
3/5, 3/6, 3/7, 3/8, ...
4/7, 4/10, 4/13, 4/16, ...
Looking at harmonic sequences that way makes each of these problems easy, because in each example the numerators of the first and last terms are the same. So inserting the specified number of harmonic means between the two given numbers simply means making the denominators an arithmetic sequence.
(1) 1/2 and 1/6; insert 3 harmonic means
The numerators are all 1; the denominators are 2, __, __, __, 6. That's easy -- the denominators are 2, 3, 4, 5, and 6.
ANSWER: 1/2, 1/3, 1/4, 1/5, 1/6
(2) 2/3 and 2/9; insert 2 harmonic means
The numerators are all 2. The denominators 3, a, b, 9 form an arithmetic sequence; so the common difference between the denominators has to be 2.
ANSWER: 2/3, 2/5, 2/7, 2/9
(3) 5/2 and 5/27; insert 4 harmonic means
The numerators are all 5. The difference 27-2=25 needs to be broken into 4+1=5 equal parts; 25/5=5. So the common difference between the denominators is 5.
ANSWER: 5/2, 5/7, 5/12, 5/17, 5/22, 5/27
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