Given questions are based on a missing series pattern and following that pattern find the relation between P,Q and R.
`35," "70," "210," "P," "4200," "25200`
`140," "280," "Q," "3360," "16800," "100800`
`40," "80," "240," "R," "4800," "28800`

Options

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

Answer

To solve the problem, we need to identify the pattern in each of the three series and find the values of P, Q, and R. Let's break it down step by step.

### Step 1: Analyze the first series
The first series is:
- 35, 70, 210, P, 4200, 25200

To find the pattern:
- From 35 to 70: \( 35 \times 2 = 70 \)
- From 70 to 210: \( 70 \times 3 = 210 \)
- From 210 to P: The next multiplier should be 4.
- From P to 4200: The multiplier from P to 4200 should be 5.
- From 4200 to 25200: \( 4200 \times 6 = 25200 \)

Now, let's calculate P:
- \( P = 210 \times 4 = 840 \)

### Step 2: Analyze the second series
The second series is:
- 140, 280, Q, 3360, 16800, 100800

To find the pattern:
- From 140 to 280: \( 140 \times 2 = 280 \)
- From 280 to Q: The next multiplier should be 3.
- From Q to 3360: The multiplier from Q to 3360 should be 4.
- From 3360 to 16800: \( 3360 \times 5 = 16800 \)
- From 16800 to 100800: \( 16800 \times 6 = 100800 \)

Now, let's calculate Q:
- \( Q = 280 \times 3 = 840 \)

### Step 3: Analyze the third series
The third series is:
- 40, 80, 240, R, 4800, 28800

To find the pattern:
- From 40 to 80: \( 40 \times 2 = 80 \)
- From 80 to 240: \( 80 \times 3 = 240 \)
- From 240 to R: The next multiplier should be 4.
- From R to 4800: The multiplier from R to 4800 should be 5.
- From 4800 to 28800: \( 4800 \times 6 = 28800 \)

Now, let's calculate R:
- \( R = 240 \times 4 = 960 \)

### Step 4: Establish the relationship between P, Q, and R
Now we have:
- \( P = 840 \)
- \( Q = 840 \)
- \( R = 960 \)

The relationship can be expressed as:
- \( P = Q \) and both \( P \) and \( Q \) are less than \( R \).

### Final Answer
Thus, the relation between P, Q, and R is:
- \( P = Q < R \)