Let P(n) be the statement "7 divides `2^(3n)-1`." What is `P(n+1)`?

Answer

To find \( P(n+1) \) based on the statement \( P(n) \), we start with the original statement:

1. **Identify the original statement**:
The statement \( P(n) \) is defined as "7 divides \( 2^{3n} - 1 \)". This can be mathematically expressed as:
\[ P(n): 7 \mid (2^{3n} - 1) \]

2. **Substituting \( n \) with \( n+1 \)**:
To find \( P(n+1) \), we replace \( n \) in the original statement with \( n+1 \). Thus, we have:
\[ P(n+1): 7 \mid (2^{3(n+1)} - 1) \]

3. **Simplifying the expression**:
We can simplify \( 3(n+1) \) as follows:
\[ 3(n+1) = 3n + 3 \]
Therefore, we can rewrite \( P(n+1) \) as:
\[ P(n+1): 7 \mid (2^{3n + 3} - 1) \]

4. **Final expression**:
We can express \( 2^{3n + 3} \) in terms of \( 2^{3n} \):
\[ 2^{3n + 3} = 2^{3n} \cdot 2^3 = 2^{3n} \cdot 8 \]
Thus, we can write:
\[ P(n+1): 7 \mid (8 \cdot 2^{3n} - 1) \]

5. **Conclusion**:
The final statement for \( P(n+1) \) is:
\[ P(n+1): 7 \mid (2^{3n + 3} - 1) \]