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Question
If u, v, w, and x are in continued proportion, then prove that (2u+3x) : (3u+4x) : : (2u3+3v3) : (3u3+4v3)
`("pqr")^2 (1/"p"^4 + 1/"q"^4 + 1/"r"^4) = ("p"^4 + "q"^4 + "r"^4)/"q"^2`
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Solution
p : q : : q : r ⇒ q2 = pr
`("pqr")^2 (1/"p"^4 + 1/"q"^4 + 1/"r"^4) = ("p"^4 + "q"^4 + "r"^4)/"q"^2`
LHS
`("pqr")^2 (1/"p"^4 + 1/"q"^4 + 1/"r"^4)`
`= ("q" xx "q"^2)^2 (("q"^4"r"^4 + "p"^4"r"^4 + "p"^4"q"^4)/("p"^4"q"^4"r"^4))`
`= "q"^6 (("q"^4"r"^4 + "q"^8 + "p"^4"q"^4)/("q"^8"q"^4))`
`= "q"^6 (("r"^4 + "q"^4 + "p"^4)/"q"^8)`
`= (("r"^4 + "q"^4 + "p"^4)/"q"^2)` = RHS
LHS = RHS , Hence Proved.
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