In an arithmetic sequence, the general formula for the $n$-th term is:

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

where:

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

Step 1: Find the common difference $d$

We are given:

• $a_1 = 7$,
• $a_2 = 2$.

The common difference $d$ can be calculated as:

$d = a_2 - a_1 = 2 - 7 = -5$

Step 2: Use the formula to find $n$ when $a_n = -18$

We now want to find which term $n$ gives $a_n = -18$. Using the general formula for $a_n$:

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

Substitute the known values:

$-18 = 7 + (n-1) \cdot (-5)$

Step 3: Solve for $n$

Simplify the equation:

$-18 = 7 - 5(n-1)$

$-18 = 7 - 5n + 5$

$-18 = 12 - 5n$

$-18 - 12 = -5n$

$-30 = -5n$

$n = \frac{30}{5} = 6$

Thus, the $6$-th term of the arithmetic sequence is $-18$.

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Would you like further details or have any questions?

Here are 5 related questions for you:

1. How do you calculate the sum of the first $n$ terms of an arithmetic sequence?
2. What is the formula for the common difference in an arithmetic sequence?
3. How can we find the $n$-th term if the sequence has a positive common difference?
4. Can you explain how an arithmetic sequence differs from a geometric sequence?
5. How does knowing two terms in an arithmetic sequence help in finding others?

Tip: When solving for unknowns in arithmetic sequences, identifying the common difference is crucial—it drives how fast the terms grow or shrink.