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Question
If p, q, r ands are In continued proportion, then prove that (p3+q3+r3) ( q3+r3+s3) : : P : s
Solution
`"p"/"q" = "q"/"r" = "r"/"s" = "k"`
r = ks
q = kr = k2s
p = kq = k3s
LHS
`("p"^3 + "q"^3 + "r"^3)/("q"^3 + "r"^3 + "s"^3)`
`= ("k"^9"s"^3 + "k"^6 "s"^3 + "k"^3"s"^3)/("k"^6"s"^3 + "k"^3"s"^3 + "s"^3)`
`= ("s"^3"k"^3("k"^6 + "k"^3 + 1))/("s"^3("k"^6 + "k"^3"s"^3 + "s"^3))`
`= "k"^3`
RHS
`"p"/"s" = ("k"^3"s")/"s" = "k"^3`
LHS = RHS . Hence proved.
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