If `P = (101)^(100) and Q = (100)^(101)`, then the correct relation is:

Options

A. `P gt Q`
B. `P lt Q`
C. `P = Q`
D. `P = 11/10 Q`

Answer

To determine the relationship between \( P = 101^{100} \) and \( Q = 100^{101} \), we can analyze the two expressions step by step.

### Step 1: Rewrite the expressions
We start with the definitions:
- \( P = 101^{100} \)
- \( Q = 100^{101} \)

### Step 2: Express \( Q \) in terms of \( P \)
We can rewrite \( Q \) as follows:
\[ Q = 100^{101} = (100^{100}) \cdot (100) = 100^{100} \cdot 100 \]

### Step 3: Compare \( P \) and \( Q \)
Next, we can express \( P \) in a similar form:
\[ P = 101^{100} = (100 + 1)^{100} \]
Using the Binomial Theorem, we can expand \( (100 + 1)^{100} \):
\[ (100 + 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} 100^{100-k} \cdot 1^k \]
The first term of this expansion is \( 100^{100} \), and the next term is \( \binom{100}{1} \cdot 100^{99} \cdot 1 = 100 \cdot 100^{99} = 100^{100} \).

### Step 4: Analyze the leading terms
The leading term of \( P \) is \( 100^{100} \), and the next term contributes positively to \( P \):
\[ P = 100^{100} + \text{(other positive terms)} \]
Thus, we can conclude that:
\[ P > 100^{100} \]

### Step 5: Compare \( P \) and \( Q \)
Now, we have:
- \( P = 100^{100} + \text{(positive terms)} \)
- \( Q = 100^{100} \cdot 100 \)

Since \( 100^{100} \) is a common factor, we can compare:
\[ P = 100^{100} + \text{(positive terms)} \quad \text{and} \quad Q = 100^{101} \]
Since \( Q \) is \( 100^{100} \cdot 100 \), we can see that:
\[ P < Q \]

### Conclusion
Thus, the correct relation is:
\[ Q > P \]