Finding $c$ given $a, ab, abc$ are in A.P

Question

If a, b, c are distinct nonzero integers such that a, ab, abc are in A.P then find the value of c. The answer is one among this : 1, 2, 3 or 4.

I tried $$\frac{a+abc}{2}=ab$$ But solving doesn't seem to give a numerical answer independent of b.

Answer 1

Your equation becomes $1+bc=2b$ If $b \neq \pm 1$ the right side is divisible by $b$ and the left is not, so those are the only choices for $b$. We can just try them both. If $b=1$ we have $1+c=2$, which requires $c=1$ but $b,c$ are distinct, so $b=-1, 1-c=-2, c=3$. We cannot find $a$, but were not asked to do so.

Answer 2

Multiplying by $2/(ab)$ gives $$\frac1b+ c=2$$ for this to be an integer $b$ had better be $1$ or $-1$ etc.