if " a (1/b + 1/c) , b (1/c + 1/a) , c (1/a + 1/b) " are in A.P. prove a, b, c are in A.P.?

Answer 1

a(1/b+1/c) + c(1/a+1/b) = 2b(1/c+1/a)

so a b and c all are non zero

add 2 on both sides

a(1/b+1/c) +1 + c(1/a+1/b) + 1 = 2b(1/c+1/a) + 2
or a(1/b+1/c+1/a) + c(1/a+1/b+1/c) = 2b(1/c+1/a+1/b)

devide by 1/b+1/c+1/a to get

a+c = 2b or a b and c are in AP