The given image asks for the formula of the $n$-th term of an arithmetic sequence with the terms: $-18, -15, -12, -9, \dots$.

Step 1: Find the Common Difference

The common difference $d$ can be found by subtracting any term from the next one: $d = -15 - (-18) = -15 + 18 = 3$

Step 2: Use the Formula for the $n$-th Term

The general formula for the $n$-th term of an arithmetic sequence is: $U_n = U_1 + (n-1) \cdot d$ where $U_1$ is the first term and $d$ is the common difference.

Here, $U_1 = -18$ and $d = 3$. Substituting these values into the formula: $U_n = -18 + (n-1) \cdot 3$ Simplifying: $U_n = -18 + 3n - 3$ $U_n = 3n - 21$

Answer:

The correct formula is $U_n = 3n - 21$, which corresponds to option e.

Would you like more details or have any further questions? Here are 5 related questions to explore:

1. How do we derive the formula for the sum of an arithmetic series?
2. What happens if the common difference in an arithmetic sequence is negative?
3. How can you determine the number of terms in a given arithmetic sequence?
4. How do you find the middle term of an arithmetic sequence with an odd number of terms?
5. Can an arithmetic sequence have a fractional common difference?

Tip: In an arithmetic sequence, the common difference remains constant between consecutive terms, making it easy to predict future terms.