Let P (n) be the statement." `2^(3n-1)` is integral multiple of 7". Then, P (1), P (2) and P (3) are true ?

Answer

To determine whether the statements P(1), P(2), and P(3) are true, we will evaluate each case step by step.

### Step 1: Evaluate P(1)
We need to check if \( P(1) \) is true, which means we need to evaluate \( 2^{3(1) - 1} \).

\[ P(1) = 2^{3 \cdot 1 - 1} = 2^{3 - 1} = 2^2 = 4 \]

Now we check if 4 is an integral multiple of 7. Since 4 is not divisible by 7, we conclude that:

\[ P(1) \text{ is FALSE.} \]

### Step 2: Evaluate P(2)
Next, we check \( P(2) \):

\[ P(2) = 2^{3(2) - 1} = 2^{6 - 1} = 2^5 = 32 \]

Now we check if 32 is an integral multiple of 7. Since 32 is not divisible by 7, we conclude that:

\[ P(2) \text{ is FALSE.} \]

### Step 3: Evaluate P(3)
Finally, we check \( P(3) \):

\[ P(3) = 2^{3(3) - 1} = 2^{9 - 1} = 2^8 = 256 \]

Now we check if 256 is an integral multiple of 7. Since 256 is not divisible by 7, we conclude that:

\[ P(3) \text{ is FALSE.} \]

### Conclusion
Since \( P(1) \), \( P(2) \), and \( P(3) \) are all FALSE, we can conclude that the statement "P(1), P(2), and P(3) are true" is incorrect.

### Summary
- \( P(1) \) is FALSE.
- \( P(2) \) is FALSE.
- \( P(3) \) is FALSE.