To solve this problem, let’s first analyze the given sequence and then find the equation for the $n$-th term as well as the 50th term, $a_{50}$.

Step 1: Identify the First Term and Common Difference

The sequence given is: $-29, -12, 5, 22, \ldots$

1. First term ($a_1$): The first term is $-29$.

2. Common difference ($d$): We find the common difference by subtracting consecutive terms.

   $d = -12 - (-29) = -12 + 29 = 17$

So, the common difference $d = 17$.

Step 2: Write the General Formula for the $n$-th Term

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

Substitute $a_1 = -29$ and $d = 17$: $a_n = -29 + (n - 1) \times 17$ Simplify this expression: $a_n = -29 + 17n - 17$ $a_n = 17n - 46$

So, the formula for the $n$-th term is: $a_n = 17n - 46$

Step 3: Find $a_{50}$

Now, substitute $n = 50$ into the formula to find $a_{50}$: $a_{50} = 17 \times 50 - 46$ $a_{50} = 850 - 46$ $a_{50} = 804$

Final Answers

• The $n$-th term formula: $a_n = 17n - 46$
• The 50th term: $a_{50} = 804$

Would you like further details or explanations on any part?

Here are five additional questions to deepen your understanding:

1. How can you find the sum of the first 50 terms of this arithmetic sequence?
2. What would the 100th term be in this sequence?
3. How does the general formula for the $n$-th term change if the common difference is negative?
4. How can you check if a number, like 500, is part of this sequence?
5. How would you derive the common difference if only the 1st and 3rd terms are given?

Tip: Always verify the common difference by checking multiple consecutive pairs of terms to ensure the sequence is arithmetic.