4,2,2,3,6,15,45
If `(2700)^(P)` is the term of the sequence where P is the `P^(th)` term of sequence then find 'P'.

Options

A. 8th
B. 9th
C. 10th
D. 11th

Explanation

To solve the problem step by step, we need to identify the pattern in the given sequence and find the term that corresponds to \( (2700)^P \).

### Step 1: Identify the Sequence
The sequence given is:
\[ 4, 2, 2, 3, 6, 15, 45 \]

### Step 2: Analyze the Pattern
Let's look at how each term relates to the previous one:
- The first term is \( 4 \).
- The second term is \( 2 \) which is \( 4 \times 0.5 \).
- The third term is \( 2 \) which is \( 2 \times 1 \).
- The fourth term is \( 3 \) which is \( 2 \times 1.5 \).
- The fifth term is \( 6 \) which is \( 3 \times 2 \).
- The sixth term is \( 15 \) which is \( 6 \times 2.5 \).
- The seventh term is \( 45 \) which is \( 15 \times 3 \).

From this, we can see that the multipliers are:
- \( 0.5, 1, 1.5, 2, 2.5, 3 \)

### Step 3: Continue the Pattern
Following the pattern, the next multiplier should be \( 3.5 \):
- The eighth term would be \( 45 \times 3.5 = 157.5 \).

Continuing this, the ninth term would be:
- The ninth term would be \( 157.5 \times 4 = 630 \).

The tenth term would be:
- The tenth term would be \( 630 \times 4.5 = 2835 \).

### Step 4: Find \( P \) such that \( (2700)^P \) is a term
We need to find \( P \) such that \( (2700)^P \) is one of the terms in the sequence. The terms we found are:
1. \( 4 \)
2. \( 2 \)
3. \( 2 \)
4. \( 3 \)
5. \( 6 \)
6. \( 15 \)
7. \( 45 \)
8. \( 157.5 \)
9. \( 630 \)
10. \( 2835 \)

### Step 5: Compare with \( (2700)^P \)
Now, we need to check which of these terms can be expressed as \( (2700)^P \):
- \( 2700 = 27 \times 100 = 3^3 \times 10^2 = 3^3 \times (10^2) \)
- \( 2700^1 = 2700 \)
- \( 2700^0 = 1 \)

None of the terms directly match \( 2700 \) or \( 2700^1 \), but we can see that \( 630 \) is close to \( 2700 \) and \( 2835 \) is also close.

### Conclusion
However, since \( 2835 \) is the tenth term and \( 2700 \) is not directly a term, we can conclude that:
- The closest term to \( 2700 \) in the sequence is \( 2835 \), which is the 10th term.

Thus, \( P = 10 \).

### Final Answer
The value of \( P \) is \( 10 \).

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