Given questions are based on a missing series pattern and following that pattern find the relation between P,Q and R.
`2700," "5400," "P," "7200," "1440," "8640`
`2100," "Q," "1400," "5600," "1120," "6720`
`1500," "3000," "1000," "R," "800," "4800`

Options

A. `P gt Q gt R`
B. `P lt Q = R`
C. `P lt Q gt R`
D. `P = Q gt R`

Answer

To solve the problem, we need to analyze the three given series and identify the missing terms \( P, Q, \) and \( R \) based on the patterns in the series.

### Step-by-Step Solution:

1. **Identify the First Series:**
The first series is:
\[ 2700, 5400, P, 7200, 1440, 8640 \]
- From \( 2700 \) to \( 5400 \):
\[ 2700 \times 2 = 5400 \]
- From \( 5400 \) to \( P \):
- We will find \( P \) later.
- From \( P \) to \( 7200 \):
- From \( 7200 \) to \( 1440 \):
\[ 7200 \div 5 = 1440 \]
- From \( 1440 \) to \( 8640 \):
\[ 1440 \times 6 = 8640 \]

**Pattern Identified:**
The pattern alternates between multiplication and division:
- Multiply by 2, then divide by 5, then multiply by 6.
- Therefore, we can deduce:
\[ P = 5400 \div 3 = 1800 \]

2. **Identify the Second Series:**
The second series is:
\[ 2100, Q, 1400, 5600, 1120, 6720 \]
- From \( 2100 \) to \( Q \):
\[ 2100 \times 2 = 4200 \]
- From \( Q \) to \( 1400 \):
\[ 4200 \div 3 = 1400 \]
- From \( 1400 \) to \( 5600 \):
\[ 1400 \times 4 = 5600 \]
- From \( 5600 \) to \( 1120 \):
\[ 5600 \div 5 = 1120 \]
- From \( 1120 \) to \( 6720 \):
\[ 1120 \times 6 = 6720 \]

**Value of \( Q \):**
\[ Q = 4200 \]

3. **Identify the Third Series:**
The third series is:
\[ 1500, 3000, 1000, R, 800, 4800 \]
- From \( 1500 \) to \( 3000 \):
\[ 1500 \times 2 = 3000 \]
- From \( 3000 \) to \( 1000 \):
\[ 3000 \div 3 = 1000 \]
- From \( 1000 \) to \( R \):
\[ 1000 \times 4 = 4000 \]
- From \( R \) to \( 800 \):
\[ 4000 \div 5 = 800 \]
- From \( 800 \) to \( 4800 \):
\[ 800 \times 6 = 4800 \]

**Value of \( R \):**
\[ R = 4000 \]

4. **Final Values:**
- \( P = 1800 \)
- \( Q = 4200 \)
- \( R = 4000 \)

5. **Determine the Relationship:**
- \( P < Q \) (1800 < 4200)
- \( Q > R \) (4200 > 4000)

### Conclusion:
The relationship between \( P, Q, \) and \( R \) is:
\[ P < Q > R \]