Question

If a $b^2$$c^3$, $a^2$$b^3$$c^4$, $a^3$$b^4$$c^5$ are in A.P. (a, b, c > 0) then the minimum value of a + b + c is __

Answer 1

Observe that a + b + c is a part of AM of a, b, c i.e.,$\frac{(a+b+c)}{3}$ as AM $\ge$ GM & HM,

We can get minimum value of AM from this relation : AM $\ge$ GM

Using AM $\ge$ GM $\Rightarrow$ a + b + c $\ge$ 3 $(abc{)}^{\frac{1}{3}}$ --------------(1)

We need value of abc

Given a $b^2$ $c^3$, $a^2$ $b^3$ $c^4$, $a^3$ $b^4$ $c^5$ are in A.P.

$\frac{2a^2{b}^3{c}^4}{a{b}^2{c}^3}$ = $\frac{a{b}^2{c}^3}{a{b}^2{c}^3}$ + $\frac{a^3{b}^4{c}^5}{a{b}^2{c}^3}$

$\Rightarrow$ 2abc = 1 + $a^2$ $b^2$ $c^2$

$\Rightarrow$ ${\left(abc\right)}^2$ - 2abc + 1 = 0

$\Rightarrow$ ${(abc-1)}^2$ = 0

So, abc = 1

From (1), a + b + c $\ge$ 3

So, the minimum value is 3.