Smallest Value in Geometric Progression

Given $T_m = \frac{3}{4}$ and $T_n = 12$ with $m < n$. Then $\frac{T_n}{T_m} = r^{n-m} = \frac{12}{3/4} = 16$. Let $k = n - m > 0$. So $r^k = 16$. Since $r$ is an integer, possible values of $(r, k)$ are:

• If $k = 1$, $r = 16 \implies r + k = 16 + 1 = 17$
• If $k = 2$, $r = 4$ or $r = -4 \implies r + k = 4 + 2 = 6$ or $-4 + 2 = -2$
• If $k = 4$, $r = 2$ or $r = -2 \implies r + k = 2 + 4 = 6$ or $-2 + 4 = 2$ The smallest possible value of $r + n - m = r + k$ is $-2$.