1.

If ` x : a = y : b = z : c `, then prove that `x^(3)/a^(2) + y^(3)/b^(2) + z^(3)/c^(2) = ((x+y+z)^(3))/((a+b+c)^(2))`

Answer» Given that ` x : a = y : b = z : c`
or, ` x/a = y/b = z/c = k (k ne 0) `(let)
` :. X = ak, y = bk , z = ck`.
LHS `= x^(3)/a^(2) + y^(3)/b^(2) + z^(3)/c^(2) = ((ak)^(3))/a^(2) + ((bk)^(3))/b^(2) + ((ck)^(3))/c^(2)`
` = (a^(3)k^(3))/a^(2) + (b^(3) k^(3))/b^(2) + (c^(3)k^(3))/c^(2) = ak^(3) + bk^(3) + ck^(3)`
` = k^(3) (a + b + c)`.
RHS ` = ((x+y+z)^(3))/((a+b+c)^(2)) = ((ak+bk+ck)^(3))/((a+b+c)^(2))`
` = ({k(a+b+c)}^(3))/((a+b+c)^(2))= (k^(3)(a+b+c)^(3))/((a+b+c)^(2)) = k^(3)(a+b+c)`.
` :. ` LHS = RHS


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