We are given an arithmetic progression (AP) with the first few terms: $-8, -5, \ldots, 49$. The goal is to find the 7th term of this AP.

Step 1: Identify the first term $a_1$ and the common difference $d$.

From the given sequence:

• The first term $a_1 = -8$.
• The second term is $-5$, so the common difference $d$ can be calculated as: $d = a_2 - a_1 = -5 - (-8) = -5 + 8 = 3$

Step 2: Use the formula for the $n$-th term of an AP.

The general formula for the $n$-th term of an AP is: $a_n = a_1 + (n - 1) \cdot d$

For the 7th term ($n = 7$): $a_7 = a_1 + (7 - 1) \cdot d = -8 + 6 \cdot 3 = -8 + 18 = 10$

Thus, the 7th term of the AP is 10.

Conclusion:

The 7th term of the arithmetic progression is 10.

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Do you need further details or clarifications? If you'd like to dive deeper, here are some related questions:

1. How would you find the 15th term of this arithmetic progression?
2. What is the sum of the first 10 terms of this AP?
3. How do you determine the common difference $d$ if only a few terms of an AP are given?
4. How do you identify whether a given sequence is an arithmetic progression?
5. What is the general formula for the sum of the first $n$ terms of an arithmetic progression?

Tip: To confirm that a sequence is an AP, check if the difference between consecutive terms is constant. If it is, you've found an arithmetic progression!