The sum of the first $n$ elements of an arithmetic sequence is $A$, and the sum of the first $2n$ elements is $B$. Express the sum of the first $3n$ elements using $A$ and $B$.

Express the sum of the first $3n$ elements using $A$ and $B$.
Let's write down the known sum formula for the arithmetic sequence for the first $n$, $2n$, and $3n$ terms:

$$
A=S_{n}=\frac{n}{2}\left(2 a_{1}+(n-1) d\right), B=S_{2 n}=\frac{2 n}{2}\left(2 a_{1}+(2 n-1) d\right), C=S_{3 n}=\frac{3 n}{2}\left(2 a_{1}+(3 n-1) d\right)
$$

Rearranging:

$$
2 A=2 a_{1} n+n^{2} d-n d, B=2 a_{1} n+2 n^{2} d-n d
$$

Subtracting the first equation from the second:

$$
B-2 A=n^{2} d
$$

Rearranging the third equation and substituting $n^{2} d$ with the obtained $(B-2 A)$:

$$
2 C=2 S_{3 n}=6 a_{1} n+9 n^{2} d-3 n d=6 a_{1} n+6 n^{2} d-3 n d+3 n^{2} d=3\left(2 a_{1} n+2 n^{2} d-n d\right)+3 n^{2} d=3 B+3(B-2 A)
$$

From this, we get $C=3(B-A)$.