Which number is larger `(1.3)^(2000) or 600`

Answer

To determine which number is larger between \( (1.3)^{2000} \) and \( 600 \), we can follow these steps:

### Step 1: Rewrite \( (1.3)^{2000} \)
We can express \( 1.3 \) as \( 1 + 0.3 \). Therefore, we can rewrite \( (1.3)^{2000} \) as:
\[ (1.3)^{2000} = (1 + 0.3)^{2000} \]

### Step 2: Use the Binomial Theorem
We can apply the Binomial Theorem to expand \( (1 + 0.3)^{2000} \):
\[ (1 + 0.3)^{2000} = \sum_{k=0}^{2000} \binom{2000}{k} (0.3)^k \]
This expansion gives us:
\[ = \binom{2000}{0} (0.3)^0 + \binom{2000}{1} (0.3)^1 + \binom{2000}{2} (0.3)^2 + \ldots + \binom{2000}{2000} (0.3)^{2000} \]

### Step 3: Calculate the First Few Terms
Calculating the first few terms of the expansion:
- The first term is \( \binom{2000}{0} (0.3)^0 = 1 \).
- The second term is \( \binom{2000}{1} (0.3)^1 = 2000 \times 0.3 = 600 \).
- The third term is \( \binom{2000}{2} (0.3)^2 = \frac{2000 \times 199}{2} \times (0.3)^2 = \frac{2000 \times 199}{2} \times 0.09 = 199 \times 300 = 59700 \).

### Step 4: Sum the Terms
Now, we can sum the first few terms:
\[ (1.3)^{2000} \approx 1 + 600 + 59700 + \ldots \]
Clearly, the sum of these terms is significantly greater than \( 600 \).

### Step 5: Conclusion
Since the expansion of \( (1.3)^{2000} \) gives us a sum that is greater than \( 600 \), we conclude that:
\[ (1.3)^{2000} > 600 \]

Thus, \( (1.3)^{2000} \) is larger than \( 600 \).

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