Find the 18th term of the series 24,18,12

To find the 18th term of the series 24, 18, 12, we first need to determine whether this series is arithmetic or geometric.

### Step 1: Identify the type of series

- **Arithmetic Series**: The difference between consecutive terms is constant.
- **Geometric Series**: The ratio between consecutive terms is constant.

Let's check the differences and the ratios between consecutive terms in the given series:

1. **Difference between terms**:
   - \(18 - 24 = -6\)
   - \(12 - 18 = -6\)

   The difference between consecutive terms is \(-6\), which is constant.

2. **Ratio between terms**:
   - \(\frac{18}{24} = \frac{3}{4}\)
   - \(\frac{12}{18} = \frac{2}{3}\)

   The ratio between consecutive terms is not constant.

Since the difference between consecutive terms is constant, this is an **arithmetic series** with a common difference of \(-6\).

### Step 2: Find the 18th term of the arithmetic series

The formula for the \(n\)-th term (\(a_n\)) of an arithmetic series is:

\[
a_n = a_1 + (n - 1) \times d
\]

where:
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

For this series:
- \(a_1 = 24\)
- \(d = -6\)
- \(n = 18\)

Plugging these values into the formula:

\[
a_{18} = 24 + (18 - 1) \times (-6)
\]

\[
a_{18} = 24 + 17 \times (-6)
\]

\[
a_{18} = 24 - 102
\]

\[
a_{18} = -78
\]

### Conclusion

The 18th term of the series 24, 18, 12 is \(-78\).

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Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the sum of the first 18 terms of this arithmetic series?
2. What is the 25th term of this series?
3. How do you determine if a series is arithmetic or geometric?
4. Can a series be both arithmetic and geometric?
5. How does the common difference affect the nature of an arithmetic series?

**Tip:** In an arithmetic series, each term is obtained by adding a constant difference to the previous term.

Learn how to find the 18th term of the arithmetic series 24, 18, 12 using the nth-term formula. This problem covers arithmetic series concepts, including identifying the common difference and calculating specific terms. Suitable for students in Grades 7-9.

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