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