If the first and the third terms of a G.P. are 2 and 8 respectively, find its second term.

Answer

To solve the problem, we need to find the second term of a geometric progression (G.P.) given that the first term (T1) is 2 and the third term (T3) is 8.

### Step-by-Step Solution:

1. **Identify the first term (A) and the third term (T3)**:
- We are given that the first term \( T_1 = 2 \).
- The third term \( T_3 = 8 \).

2. **Use the formula for the n-th term of a G.P.**:
- The formula for the n-th term of a G.P. is given by:
\[ T_n = A \cdot R^{n-1} \]
- For the first term:
\[ T_1 = A \cdot R^{1-1} = A \cdot R^0 = A \]
- Therefore, we have:
\[ A = 2 \]

3. **Find the expression for the third term**:
- For the third term:
\[ T_3 = A \cdot R^{3-1} = A \cdot R^2 \]
- Substituting the known values:
\[ 8 = 2 \cdot R^2 \]

4. **Solve for the common ratio (R)**:
- Rearranging the equation:
\[ R^2 = \frac{8}{2} = 4 \]
- Taking the square root of both sides:
\[ R = 2 \]

5. **Find the second term (T2)**:
- Now, we can find the second term using the formula:
\[ T_2 = A \cdot R^{2-1} = A \cdot R \]
- Substituting the values of A and R:
\[ T_2 = 2 \cdot 2 = 4 \]

### Final Answer:
The second term of the G.P. is **4**.