Question

If `a = (b + c)/(2), c = (a + b)/(2)` and b is mean proportional between a and c, prove that `(1)/a + (1)/c = (1)/b`.

Answer 1

b is the mean proportional of a and c
b² = ac
b² = `((b + c)/(2)). ((a + b)/(2))`
⇒ 4b² = ab + ac + b² + bc  ...[∵ b² = ac]
⇒  4b² = ab + 2b² + bc
2b² - ab + bc,
`["Dividing both sides by abc"]`
⇒  `(2b^2)/(abc) = (ab)/(abc) + (bc)/(abc)`
⇒  `(2)/b = (1)/c + (1)/a`,           ...[∵ b² = ac]
Hence proved.