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 $a_n = a_1 \cdot r^{n-1}$.