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The question involves a geometric sequence where the first three terms are \( 2 \), \( 3x \), and \( x + 16 \). 

### Part (a) - Find the possible values of \( x \)

In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be \( r \).

For the terms \( 2 \), \( 3x \), and \( x + 16 \), the common ratio between the first two terms is:
\[
r = \frac{3x}{2}
\]
The common ratio between the second and third terms is:
\[
r = \frac{x + 16}{3x}
\]
Since the ratio between consecutive terms is the same, we can set these two expressions equal:
\[
\frac{3x}{2} = \frac{x + 16}{3x}
\]
Now we solve for \( x \).

Cross-multiply to eliminate the fractions:
\[
(3x)(3x) = 2(x + 16)
\]
\[
9x^2 = 2x + 32
\]
Rearrange the equation:
\[
9x^2 - 2x - 32 = 0
\]
Now, solve this quadratic equation using the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(9)(-32)}}{2(9)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 1152}}{18}
\]
\[
x = \frac{2 \pm \sqrt{1156}}{18}
\]
\[
x = \frac{2 \pm 34}{18}
\]
Thus, the two possible values for \( x \) are:
\[
x = \frac{2 + 34}{18} = \frac{36}{18} = 2
\]
or
\[
x = \frac{2 - 34}{18} = \frac{-32}{18} = -\frac{16}{9}
\]

### Part (b) - Find the possible values of the 4th term

For the fourth term, we need to multiply the third term by the common ratio \( r \).

- If \( x = 2 \):
    - First three terms are \( 2 \), \( 6 \), and \( 18 \) (since \( 2 + 16 = 18 \)).
    - The common ratio \( r = \frac{6}{2} = 3 \).
    - The fourth term is \( 18 \times 3 = 54 \).

- If \( x = -\frac{16}{9} \):
    - First three terms are \( 2 \), \( -\frac{16}{3} \), and \( \frac{128}{9} \).
    - The common ratio \( r = \frac{-\frac{16}{3}}{2} = -\frac{8}{3} \).
    - The fourth term is \( \frac{128}{9} \times -\frac{8}{3} = -\frac{1024}{27} \).

### Final Answers:
- Possible values of \( x \): \( 2 \) or \( -\frac{16}{9} \).
- Possible values of the 4th term: \( 54 \) or \( -\frac{1024}{27} \).

Would you like any more details or have any questions?

Here are some related questions:
1. How do you find the general term of a geometric sequence?
2. What is the difference between an arithmetic and a geometric sequence?
3. Can a geometric sequence have a common ratio of 0?
4. How do you find the sum of the first \( n \) terms of a geometric series?
5. What happens when the common ratio of a geometric sequence is negative?

**Tip:** In a geometric sequence, knowing the ratio allows you to quickly determine any term in the sequence using the formula \( a_n = a_1 \cdot r^{n-1} \).

The numbers 2, 3x, and (x + 16) are the first three terms of a geometric sequence. (a) Find the possible values of x (b) Find the possible values of the 4th term of the geometric sequence.

Solve for the possible values of x and the 4th term in a geometric sequence with the terms 2, 3x, and (x + 16). Learn how to use the common ratio formula and quadratic formula to find solutions. Suitable for high school students.

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