8.2.1. (12 points) The first, second, and third terms of a geometric progression are pairwise distinct and are equal to the second, fourth, and seventh terms of a certain arithmetic progression, respectively, and the product of these three numbers is 64. Find the first term of the geometric progression.
Answer: $\frac{8}{3} \approx 2.67$.

Solution. Let the first term of the geometric progression be denoted by $b$. Then the second term of the geometric progression is $b q=b+2 d$, and the third term is $b q^{2}=b+5 d$. The product of the three numbers is $(b q)^{3}=64$, from which we obtain $b q=4$. Therefore, $b=4-2 d$ and $b q^{2}=4+3 d$. Multiplying these equations, we arrive at the quadratic equation $(b q)^{2}=(4-2 d)(4+3 d) \Leftrightarrow 16=16+4 d-6 d^{2}$. Solving for $d$, we find $d=\frac{2}{3}$, and thus $b=4-2 d=\frac{8}{3}$.