If `P(n) :1+4+7…….+(3n-2)=(1)/(2)n(3n-1)`.Verify P(n) for n =1,2.

Answer

To verify the statement \( P(n) : 1 + 4 + 7 + \ldots + (3n - 2) = \frac{1}{2} n (3n - 1) \) for \( n = 1 \) and \( n = 2 \), we will follow these steps:

### Step 1: Verify for \( n = 1 \)

1. **Calculate the left-hand side (LHS)**:
- For \( n = 1 \), the series is \( 1 \) (since \( 3 \times 1 - 2 = 1 \)).
- Therefore, \( \text{LHS} = 1 \).

2. **Calculate the right-hand side (RHS)**:
- Substitute \( n = 1 \) into the right-hand side:
\[ \text{RHS} = \frac{1}{2} \times 1 \times (3 \times 1 - 1) = \frac{1}{2} \times 1 \times (3 - 1) = \frac{1}{2} \times 1 \times 2 = 1. \]

3. **Compare LHS and RHS**:
- LHS = 1 and RHS = 1.
- Since LHS = RHS, \( P(1) \) is verified.

### Step 2: Verify for \( n = 2 \)

1. **Calculate the left-hand side (LHS)**:
- For \( n = 2 \), the series is \( 1 + 4 \) (since \( 3 \times 2 - 2 = 4 \)).
- Therefore, \( \text{LHS} = 1 + 4 = 5 \).

2. **Calculate the right-hand side (RHS)**:
- Substitute \( n = 2 \) into the right-hand side:
\[ \text{RHS} = \frac{1}{2} \times 2 \times (3 \times 2 - 1) = \frac{1}{2} \times 2 \times (6 - 1) = \frac{1}{2} \times 2 \times 5 = 5. \]

3. **Compare LHS and RHS**:
- LHS = 5 and RHS = 5.
- Since LHS = RHS, \( P(2) \) is verified.

### Conclusion

We have verified \( P(n) \) for both \( n = 1 \) and \( n = 2 \). Therefore, the statement holds true for these values of \( n \).