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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`
Solution
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
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