The problem describes an arithmetic sequence with:

• First term $a = 71$
• Common difference $d = -4$
• Last term $l = -193$

Solution:

(a) Find the number of terms ($n$) in the sequence

The general formula for the $n$-th term of an arithmetic sequence is:

$a_n = a + (n-1) d$

Substituting the known values:

$-193 = 71 + (n-1)(-4)$

$-193 - 71 = (n-1)(-4)$

$-264 = (n-1)(-4)$

$n - 1 = \frac{264}{4} = 66$

$n = 67$

So, there are 67 terms in the sequence.

(b) Find the general formula for the $n$-th term ($a_n$)

Using the general formula:

$a_n = a + (n-1)d$

$a_n = 71 + (n-1)(-4)$

$a_n = 71 - 4(n-1)$

$a_n = 71 - 4n + 4$

$a_n = 75 - 4n$

Thus, the formula for the $n$-th term is:

$a_n = 75 - 4n$

Would you like a detailed explanation for any step? Here are some related questions you might find useful:

1. How do you derive the formula for the sum of an arithmetic series?
2. What happens if the common difference is positive instead of negative?
3. How can you determine if a given number belongs to an arithmetic sequence?
4. Can you find the sum of the first 67 terms in this sequence?
5. How does an arithmetic sequence relate to real-life applications?

Tip: In an arithmetic sequence, if the common difference is negative, the terms decrease over time. Always check if the sequence will reach the given last term correctly! 😊