1. The problem is to understand the formula for the sum of a sequence: $S_n = U_1 + U_2 + \cdots + U_{n-1}$. 2. Here, $S_n$ represents the sum of the first $n-1$ terms of a sequence, where $U_k$ is the $k$-th term. 3. This formula is a general way to express the sum of terms in a sequence up to the $(n-1)$-th term. 4. If the sequence is arithmetic (each term increases by a constant difference $d$), the sum can be calculated using the formula: $$S_n = \frac{(n-1)}{2} (U_1 + U_{n-1})$$ 5. If the sequence is geometric (each term is multiplied by a constant ratio $r$), the sum is: $$S_n = U_1 \frac{1-r^{n-1}}{1-r}$$ for $r \neq 1$. 6. To find the sum, you need to know the type of sequence and the values of the terms or the common difference/ratio. 7. This formula helps in adding up terms without listing all of them individually, which is useful for large $n$.