Question

If a, b, c are in A.P show that 1/ab, 1/bc, 1/ca are also in A.P.

Answer 1

since a,b,c are in AP let d be the common difference between them, so we hav
b-a=d
c-b=d

1/bc-1/ab=1/b(1/c-1/a)=(a-c)/abc=-2d/abc
1/ca-1/bc=(b-a)/abc=d/abc

which shows theyr not in AP !