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Question

If a(1b+1c),b(1c+1a) c(1a+1b) are in A.P., prove that a, b, c are in A.P.

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Solution

We have:

a(1b+1c), b(1c+1a), c(1a+1b) are in A.P.

a(b+cbc), b(a+cac), c(a+bab) are in A.P.

ab+acbc, ab+bcac, ac+bcab are in A.P.

ab+acbc+1, ab+bcac+1, ac+bcab+1 are in A.P.

[Adding 1 to each number]

ab+ac+bcbc, ab+bc+acac, ac+bc+abab are in A.P.

or 1bc, 1ac, 1ab are in A.P.

[Dividing each number by ab + bc + ac]

or abcbc, abcac, abcab are in A.P.

[Multiplying each number by abc]

or a, b, c are in A.P.


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