Question:

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

Step-by-Step Solution

Key Concept: AP condition: $2a^2b^3c^4=ab^2c^3+a^3b^4c^5\Rightarrow 2abc=1+a^2b^2c^2\Rightarrow(abc-1)^2=0\Rightarrow abc=1$.

Min $a+b+c=\mathbf{3}$.

Correct Answer: 3