1.

If ` a : b = b : c`, then prove that `a^(2)b^(2)c^(2) = (1/a^(3) + 1/b^(3) + 1/c^(3)) = a^(3) + b^(3) + c^(3)`

Answer» Given that ` a : b = b : c`
` rArr a/b = b/c = k " (let)", [k ne 0]`
` :. A = bk = ck.k = ck^(2) " and " b = ck`.
LHS ` = a^(2)b^(2)c^(2) (1/a^(3)+1/b^(3)+1/c^(3))=(b^(2)c^(2))/a + (c^(2)a^(2))/b+(a^(2)b^(2))/c`
` = ((ck)^(2).c^(2))/(ck^(2)) + (c^(2)(ck^(2))^(2))/(ck) + ((ck^(2))^(2)* (ck)^(2))/c`
` = c^(3) + c^(3) k^(3) + c^(3) k^(6) = c^(3) (1+k^(3)+k^(6))`
RHS `= a^(3) + b^(3) + c^(3) = (ck^(2))^(3) + (ck)^(3) + c^(3)`
` = c^(3) + c^(3) + c^(3) k^(6) = c^(3) (1 + k^(3) + k^(6))` ltbr. ` :. ` LHS = RHS


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