To find a formula for the $n$-th term of the sequence:

Given sequence: $-3, 5, 13, \dots$

Step 1: Identify the pattern of the sequence

Let's look at the differences between consecutive terms:

1. $5 - (-3) = 8$
2. $13 - 5 = 8$

Since the difference between each pair of terms is constant ($+8$), this sequence is an arithmetic sequence.

Step 2: Find the first term and common difference

For an arithmetic sequence, the formula for the $n$-th term is: $a_n = a + (n - 1) \cdot d$ where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term position in the sequence.

In this case:

• The first term $a = -3$,
• The common difference $d = 8$.

Step 3: Write the formula for the $n$-th term

Substitute $a = -3$ and $d = 8$ into the formula: $a_n = -3 + (n - 1) \cdot 8$

Step 4: Simplify the formula

Expanding and simplifying: $a_n = -3 + 8n - 8$ $a_n = 8n - 11$

Final Answer:

The formula for the $n$-th term of the sequence is: $a_n = 8n - 11$

Would you like a detailed explanation of how this was derived, or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How would the formula change if the common difference was negative?
2. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
3. How would you verify the formula for the $n$-th term by checking a few terms?
4. Can you find the 10th term of this sequence using the formula?
5. How can you determine if a given number is a term in this sequence?

Tip: In arithmetic sequences, each term is obtained by adding a constant value, the common difference, to the previous term.