You provided a problem involving an arithmetic sequence: $-55, -47, -39, -31, \ldots$. We need to find the explicit formula for the sequence and compute the 50th term.

Step 1: Determine the Common Difference

The common difference $d$ can be calculated by subtracting any term from the next term in the sequence: $d = -47 - (-55) = 8$

Step 2: General Formula for an Arithmetic Sequence

The $n$-th term of an arithmetic sequence can be expressed as: $a_n = a_1 + (n-1) \cdot d$ Where:

• $a_1$ is the first term of the sequence.
• $d$ is the common difference.
• $n$ is the term number.

Step 3: Explicit Formula for the Given Sequence

Given $a_1 = -55$ and $d = 8$: $a_n = -55 + (n-1) \cdot 8$ This simplifies to: $a_n = -55 + 8n - 8 = 8n - 63$

Step 4: Find the 50th Term

To find the 50th term ($a_{50}$): $a_{50} = 8(50) - 63 = 400 - 63 = 337$

Summary

The explicit formula for the sequence is $f(n) = 8n - 63$. The 50th term of this sequence is 337.

If you need more details or have any questions, please let me know! Here are some follow-up questions you might consider:

1. How would the formula change if the first term was different?
2. What if the common difference was negative?
3. Can you predict the term number where the sequence first becomes positive?
4. How would you find the sum of the first 50 terms?
5. What patterns emerge if you graph this sequence?

Tip: Always verify the common difference by checking multiple consecutive terms to avoid calculation errors.