11. Let the geometric sequence $\left\{a_{n}\right\}$ have the first term $a_{1}=a$, and common ratio $q \neq 1$. Try to find the sum of all different products of any two terms among the first $n$ terms of this sequence.
11. From the problem, we know that the sequence $\left\{a_{n}^{2}\right\}$ is a geometric sequence with the first term $a^{2}$ and common ratio $q^{2}$, so $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ $\frac{a^{2}\left(1-q^{2 n}\right)}{1-q^{2}}$. And $\left(a_{1}+a_{2}+\cdots+a_{n}\right)^2=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}+2 \sum_{i<j}^{n} a_{i} \cdot a_{j}$, also $a_{1}+a_{2}+\cdots+a_{n}=$ $\frac{a\left(1-q^{n}\right)}{1-q}$, so we get $\sum_{\substack{i, j=1 \\ i<j}}^{n} a_{i} a_{j}=\frac{a^{2} q\left(1-q^{n}\right)\left(1-q^{n-1}\right)}{(1-q)^{2}(1+q)}$. Therefore, the required sum is $\frac{a^{2} q\left(1-q^{n}\right)\left(1-q^{n-1}\right)}{(1-q)^{2}(1+q)}$.