Let's solve the sequence to find the logical values of $P$ and $Q$.

Given sequence: $28, \frac{1}{12}, \frac{1}{6}, \frac{1}{4}, \frac{1}{2}, \frac{7}{3}, P, Q$

Step 1: Analyze the Sequence Observing the sequence, the first step is to identify a pattern or a rule that changes one term to the next.

Step 2: Find a Pattern We see that the sequence transitions from integers to fractions and back. The sequence of fractions ($\frac{1}{12}$ to $\frac{1}{2}$) seems to increase by multiplying by a constant factor, and the $\frac{7}{3}$ appears to be a bridge from fractions back towards whole numbers.

We need to first determine the multiplier:

• From $\frac{1}{12}$ to $\frac{1}{6}$, the term doubles.
• From $\frac{1}{6}$ to $\frac{1}{4}$, the term multiplies by $\frac{3}{2}$.
• From $\frac{1}{4}$ to $\frac{1}{2}$, it doubles.

Let's check the multiplier consistency:

• $\frac{1}{12} \times 2 = \frac{1}{6}$
• $\frac{1}{6} \times \frac{3}{2} = \frac{1}{4}$
• $\frac{1}{4} \times 2 = \frac{1}{2}$

It appears the multiplication alternates between 2 and $\frac{3}{2}$. However, from $\frac{1}{2}$ to $\frac{7}{3}$, a different rule may apply because the nature of the sequence shifts significantly, potentially resuming integer values.

Step 3: Project the Pattern If we assume a continued multiplication but need to determine the next factors for $P$ and $Q$, let's first reverse-engineer from $\frac{7}{3}$:

• Assuming $\frac{1}{2} \times k = \frac{7}{3}$ gives $k = \frac{14}{3}$.
• If we continue the sequence for $P$ using a multiplier, the next logical step is to multiply $\frac{7}{3}$ by $2$, given the observed alternating pattern:
  • $P = \frac{7}{3} \times 2 = \frac{14}{3}$

Step 4: Calculate $Q$ If the pattern persists (multiplying by $\frac{3}{2}$ next):

• $Q = \frac{14}{3} \times \frac{3}{2} = 7$

Thus, $P = \frac{14}{3}$ and $Q = 7$.

Would you like further details on any step or have any other questions about this sequence?

Potential questions to expand your understanding:

1. Why does the sequence start with an integer and switch to fractions?
2. Could there be an alternative pattern that fits the given sequence?
3. How can we confirm that the sequence indeed alternates multipliers?
4. What would the next number be after $Q$ if the pattern continued?
5. How do sequences like this apply in real-world scenarios?

Tip: When dealing with sequences, always look for consistency in the applied rules across several terms to validate your hypothesis about the pattern.