Question

If p, q, r ands are In continued proportion, then prove that (p³+q³+r³) ( q³+r³+s³) : : P : s 

Answer 1

`"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.