Given questions are based on a missing series pattern and following that pattern find the relation between P,Q and R.
`300," "324," "384," "504," "P," "1050`
`450," "474," "Q," "654," "864," "1200`
`200," "224," "284," "404," "R," "850`

Options

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

Answer

To solve the problem, we need to find the values of P, Q, and R in the given series based on the identified patterns. Let's break it down step by step.

### Step 1: Analyze the First Series
The first series is:
- 300, 324, 384, 504, P, 1050

**Differences:**
- 324 - 300 = 24
- 384 - 324 = 60
- 504 - 384 = 120

The differences are 24, 60, and 120. We can see that these differences can be expressed as:
- 3³ - 3 = 24
- 4³ - 4 = 60
- 5³ - 5 = 120

**Next Difference:**
For the next term, we calculate:
- 6³ - 6 = 216 - 6 = 210

**Finding P:**
Now, we add this difference to the last known term:
- P = 504 + 210 = 714

### Step 2: Analyze the Second Series
The second series is:
- 450, 474, Q, 654, 864, 1200

**Differences:**
- 474 - 450 = 24
- Q - 474 = ?
- 654 - Q = ?
- 864 - 654 = 210

From the first difference, we see that:
- 3³ - 3 = 24 (same as before)

Next, we calculate:
- 4³ - 4 = 60

**Finding Q:**
So, we add 60 to 474:
- Q = 474 + 60 = 534

### Step 3: Analyze the Third Series
The third series is:
- 200, 224, 284, 404, R, 850

**Differences:**
- 224 - 200 = 24
- 284 - 224 = 60
- 404 - 284 = 120

This follows the same pattern as the previous series.

**Next Difference:**
For the next term:
- 6³ - 6 = 210

**Finding R:**
Now, we add this difference to the last known term:
- R = 404 + 210 = 614

### Step 4: Summarize the Values
Now we have:
- P = 714
- Q = 534
- R = 614

### Step 5: Determine the Relationship Between P, Q, and R
We compare the values:
- P (714) > R (614) > Q (534)

### Conclusion
Thus, the relationship can be expressed as:
- P > R > Q