Question

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

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,

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

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

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

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

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

= `(2(ab + 2ac + bc))/(ab + 2ac + bc)`

= 2