A progression has a first term of 12 and a fifth term of 18. Find the 13th term if the progression is geometric

Given a geometric progression with the first term \( a_1 = 12 \) and the fifth term \( a_5 = 18 \), we are tasked with finding the 13th term \( a_{13} \).

### Step 1: General formula for the nth term of a geometric progression

The general formula for the nth term of a geometric progression is:

\[
a_n = a_1 \cdot r^{n-1}
\]

where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the term number.

### Step 2: Express the fifth term

Using the formula for the fifth term \( a_5 \):

\[
a_5 = a_1 \cdot r^{5-1} = a_1 \cdot r^4
\]

Substituting the values \( a_5 = 18 \) and \( a_1 = 12 \):

\[
18 = 12 \cdot r^4
\]

### Step 3: Solve for the common ratio \( r \)

Now, solve for \( r \):

\[
r^4 = \frac{18}{12} = 1.5
\]

Taking the fourth root of both sides:

\[
r = \sqrt[4]{1.5}
\]

### Step 4: Calculate the 13th term

Using the formula for the 13th term:

\[
a_{13} = a_1 \cdot r^{13-1} = a_1 \cdot r^{12}
\]

Substitute \( a_1 = 12 \) and \( r = \sqrt[4]{1.5} \):

\[
a_{13} = 12 \cdot \left( \sqrt[4]{1.5} \right)^{12}
\]

Let me calculate this.The 13th term of the geometric progression is approximately \( 40.5 \).

Would you like more details on the steps, or do you have any questions?

Here are five related questions:
1. How do you find the sum of the first n terms of a geometric progression?
2. What is the common ratio in a geometric progression and how does it affect the series?
3. How do you find the nth term of an arithmetic progression?
4. What is the difference between geometric and arithmetic progressions?
5. How do you solve for the common ratio if given any two terms of a geometric progression?

**Tip:** In geometric progressions, the common ratio can be found by dividing any term by the previous term.

A progression has a first term of 12 and a fifth term of 18. Find the 13th term if the progression is geometric.

This solution explains how to find the 13th term of a geometric progression given the first term of 12 and fifth term of 18. The problem involves using the geometric progression formula and solving for the common ratio. Suitable for students in Grades 10-12.

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