1A. Determine the second term of the arithmetic progression if the sum of the first 10 terms is 300, and the first, second, and fifth terms, in that order, form a geometric progression.
Solution. From $\frac{10}{2}\left(2 a_{1}+9 d\right)=300$ we get $2 a_{1}+9 d=60 \ldots \ldots$ (5 points) On the other hand $a_{1} a_{5}=a_{2}^{2}$, i.e. $a_{1}\left(a_{1}+4 d\right)=\left(a_{1}+d\right)^{2}$ and from this $d=2 a_{1}$.

Substituting into $2 a_{1}+9 d=60$ we get $a_{1}=3$, then $d=6$. Therefore, $a_{2}=9$.