Find the number of term of the sequence 18,15½,13...,-47

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.

Learn how to find the number of terms in the arithmetic sequence 18, 15½, 13, ... -47 with a step-by-step solution. This problem involves identifying an arithmetic sequence, finding the common difference, and using the nth term formula. Suitable for high school students (Grades 9-12).

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