To determine how many terms are in the given arithmetic sequence $5, 11, 17, \ldots, 89$, we need to identify the first term, common difference, and the last term, and then use the formula for the $n$-th term of an arithmetic sequence.

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

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

In this sequence:

• The first term $a_1 = 5$,
• The common difference $d = 11 - 5 = 6$,
• The last term $a_n = 89$.

We need to find $n$ such that: $a_n = 89$

Substitute the known values into the formula: $89 = 5 + (n-1) \cdot 6$

First, subtract 5 from both sides: $89 - 5 = (n-1) \cdot 6$ $84 = (n-1) \cdot 6$

Next, divide both sides by 6: $\frac{84}{6} = n-1$ $14 = n-1$

Finally, add 1 to both sides to solve for $n$: $n = 14 + 1$ $n = 15$

Therefore, there are $15$ terms in the arithmetic sequence.

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

1. What is the sum of the terms in this arithmetic sequence?
2. How do you find the sum of an arithmetic sequence?
3. What is the 10th term in this sequence?
4. How does the formula for the $n$-th term change if the first term and common difference are different?
5. Can you derive the formula for the sum of the first $n$ terms of an arithmetic sequence?
6. How do arithmetic sequences relate to linear functions?
7. What are some real-life applications of arithmetic sequences?
8. How would the sequence change if the common difference was 5 instead of 6?

Tip: When working with arithmetic sequences, always double-check the common difference and ensure it is consistent throughout the sequence.