Find which of the following is a G.P. :
`1/3,2/3,1,(4)/(3)`, . . . . . . . . . . . . .

Answer

To determine whether the sequence \( \frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3} \) is a geometric progression (G.P.), we need to check if the ratio between consecutive terms is constant.

### Step 1: Identify the terms
Let:
- \( T_1 = \frac{1}{3} \)
- \( T_2 = \frac{2}{3} \)
- \( T_3 = 1 \)
- \( T_4 = \frac{4}{3} \)

### Step 2: Calculate the ratio between the first two terms
The ratio \( r_1 \) between the first and second terms is given by:
\[ r_1 = \frac{T_2}{T_1} = \frac{\frac{2}{3}}{\frac{1}{3}} = \frac{2}{3} \times \frac{3}{1} = 2 \]

### Step 3: Calculate the ratio between the second and third terms
The ratio \( r_2 \) between the second and third terms is given by:
\[ r_2 = \frac{T_3}{T_2} = \frac{1}{\frac{2}{3}} = 1 \times \frac{3}{2} = \frac{3}{2} \]

### Step 4: Calculate the ratio between the third and fourth terms
The ratio \( r_3 \) between the third and fourth terms is given by:
\[ r_3 = \frac{T_4}{T_3} = \frac{\frac{4}{3}}{1} = \frac{4}{3} \]

### Step 5: Compare the ratios
Now we compare the ratios:
- \( r_1 = 2 \)
- \( r_2 = \frac{3}{2} \)
- \( r_3 = \frac{4}{3} \)

Since \( r_1 \), \( r_2 \), and \( r_3 \) are not equal, the ratios are not constant.

### Conclusion
The sequence \( \frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3} \) is **not a geometric progression (G.P.)**.

---