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Question
If a, b, c are in continued proportion, prove that a : c = (a2 + b2) : (b2 + c2).
Solution
a, b and c are the continued proportion
a: b = b: c
⇒ `a/b = b/c`
⇒ b2 = ac
Now `a/c = (a^2 + b^2)/(b^2 + c^2)`
= a(b2 + c2) = c(a2 + b2)
L.H.S.
⇒ a(b2 + c2)
⇒ a(ac + c2)
⇒ ac(a + c)
R.H.S.
⇒ c(a2 + b2)
⇒ c(a2 + ac)
⇒ ac(a + c)
L.H.S. = R.H.S.
Hence proved.
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