To find the number of terms in the sequence $18, 15\frac{1}{2}, 13, \ldots, -47$, we first identify that this sequence is an arithmetic sequence.

Step 1: Identify the first term $(a)$ and the common difference $(d)$

• The first term $a = 18$
• The second term is $15\frac{1}{2} = \frac{31}{2}$
• The third term is $13$

The common difference $d$ can be calculated as: $d = \text{second term} - \text{first term} = \frac{31}{2} - 18 = \frac{31}{2} - \frac{36}{2} = \frac{-5}{2}$

Step 2: Use the formula for the $n$th term of an arithmetic sequence

The formula for the $n$th term $a_n$ of an arithmetic sequence is: $a_n = a + (n-1)d$ We need to find the term where $a_n = -47$. So: $-47 = 18 + (n-1) \cdot \left(-\frac{5}{2}\right)$

Step 3: Solve for $n$

Let's solve the equation: $-47 = 18 - \frac{5}{2} \cdot (n-1)$ Subtract 18 from both sides: $-65 = -\frac{5}{2} \cdot (n-1)$ Multiply both sides by $-\frac{2}{5}$ to isolate $n-1$: $n-1 = \frac{2 \cdot 65}{5} = \frac{130}{5} = 26$ Thus, $n = 26 + 1 = 27$.

Final Answer:

The term $-47$ is the 27th term of the sequence.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:

1. How do you find the sum of the first $n$ terms in an arithmetic sequence?
2. What are the conditions for a sequence to be arithmetic?
3. How do you find the common difference if only two non-consecutive terms are given?
4. Can the common difference in an arithmetic sequence be zero?
5. What is the formula for the sum of an arithmetic series?

Tip: Always double-check whether a sequence is arithmetic by verifying that the difference between consecutive terms is constant.