To find the 93rd term of an arithmetic sequence, we can use the formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n - 1) \cdot d$

where:

• $a_1$ is the first term of the sequence,
• $d$ is the common difference, and
• $n$ is the term number we want to find.

Step-by-Step Solution

1. Identify the first term ($a_1$):

   • The first term, $a_1$, is given as $-5$.

2. Determine the common difference ($d$):

   • The difference between consecutive terms is $-15 - (-5) = -10$.
   • So, $d = -10$.

3. Plug the values into the formula:

   • We need to find the 93rd term, so $n = 93$.

$a_{93} = -5 + (93 - 1) \cdot (-10)$

4. Simplify the expression:
   • $93 - 1 = 92$
   • $92 \cdot (-10) = -920$
   • $a_{93} = -5 + (-920)$
   • $a_{93} = -925$

Answer

The 93rd term of the sequence is $-925$.

Would you like more details on any part of this solution?

Here are some related questions to expand your understanding:

1. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
2. How would you find the common difference if only given two non-consecutive terms?
3. Can an arithmetic sequence have a positive common difference and still produce negative terms?
4. How can we determine if a sequence is arithmetic by looking at a few terms?
5. What changes in the formula if we start counting terms from a different point?

Tip: Always verify the common difference by checking a few terms to ensure consistency.