Question

If a, b, c are in G.P. and a, x, b, y, c are in A.P., prove that `1/x + 1/y = 2/b`

Answer 1

a, b and c are in G.P.

`=>` b² = ac

a, x, b, y and c are in A.P.

`=>` 2x = a + b `=> x = (a + b)/2`

2b = x + y `=> b = (x + y)/2`

2y = b + c `=> y = (b + c)/2`

Now,

`1/x + 1/y = 2/(a + b) + 2/(b + c)`

= `(2b + 2c + 2a + 2b)/(ab + ac + b^2 + bc)`

= `(2a + 2c + 4b)/(ab + b^2 + b^2 + bc)`

= `(2a + 2c + 4b)/(ab + 2b^2 + bc)`

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

= `2/b`