To find the 515th term of the arithmetic sequence $29, 9, -11, \dots$, we need to use the formula for the $n$-th term of an arithmetic sequence:

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

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

Step 1: Find the common difference $d$

The common difference $d$ is the difference between consecutive terms. For this sequence:

$d = 9 - 29 = -20$

Step 2: Apply the formula

We know:

• $a_1 = 29$,
• $d = -20$, and
• $n = 515$.

Now, plug these into the formula:

$a_{515} = 29 + (515 - 1) \cdot (-20)$

$a_{515} = 29 + 514 \cdot (-20)$

$a_{515} = 29 - 10280$

$a_{515} = -10251$

Final Answer:

The 515th term of the sequence is $-10251$.

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

1. How would the process change if the common difference was positive?
2. Can you explain why the arithmetic sequence formula works?
3. How would you find the sum of the first 515 terms of this sequence?
4. What is the formula to find the $n$-th term of a geometric sequence?
5. How does this arithmetic sequence compare to a geometric sequence with a constant ratio?

Tip: Always check the common difference twice to avoid errors when calculating terms in arithmetic sequences.