Question

If a, b, c, d are in continued proportion, prove that:
`sqrt(ab) - sqrt(bc) + sqrt(cd) = sqrt((a - b + c) (b - c + d)`

Answer 1

Since a, b, c, d are in continued proportion then
`a/b = b/c = c/d = k`
⇒ a = bk, b = ck, c = dk
⇒ a = ck²
⇒ a = dk³, b = dk² and c = dk
L.H.S.
= `sqrt(ab) - sqrt(bc) + sqrt(cd)`
= `sqrt(dk^3·dk^2) - sqrt(dk^2·dk) + sqrt(dk·d)`
= `d·k^2 sqrt(k) - dk sqrt(k) + d sqrt(k)`
= `(k^2 - k + 1) d sqrt(k)`.
R.H.S.
= `sqrt((a - b + c)(b - c + d)`
= `sqrt((dk^3 - dk^2 + dk)(dk^2 - dk +d)`
= `sqrt(d xx d xx k(k^2 - k + 1)(k^2 - k + 1)`
= `(k^2 - k + 1)dsqrt(k)`
L.H.S. = R.H.S.
Hence proved.