Check whether the given sequences form a G.P. or not :
9,12, 16,24, . . . . . . . . . . . .

Answer

To determine whether the given sequence \(9, 12, 16, 24, \ldots\) forms a geometric progression (G.P.), we will check if the common ratio between consecutive terms is constant.

### Step-by-Step Solution:

1. **Identify the Terms**:
The given sequence is \(9, 12, 16, 24\).

2. **Calculate the Common Ratio (R)**:
The common ratio \(R\) for a G.P. is defined as the ratio of any term to the previous term. We will calculate \(R\) for the first three pairs of terms.

- **Between the 1st and 2nd terms**:
\[ R_1 = \frac{T_2}{T_1} = \frac{12}{9} = \frac{4}{3} \]

- **Between the 2nd and 3rd terms**:
\[ R_2 = \frac{T_3}{T_2} = \frac{16}{12} = \frac{4}{3} \]

- **Between the 3rd and 4th terms**:
\[ R_3 = \frac{T_4}{T_3} = \frac{24}{16} = \frac{3}{2} \]

3. **Compare the Common Ratios**:
Now we compare the calculated common ratios:
- \(R_1 = \frac{4}{3}\)
- \(R_2 = \frac{4}{3}\)
- \(R_3 = \frac{3}{2}\)

Since \(R_1\) and \(R_2\) are equal, but \(R_3\) is different, we conclude that the common ratio is not the same for all pairs of terms.

4. **Conclusion**:
Since the common ratio is not constant, the sequence \(9, 12, 16, 24, \ldots\) does not form a geometric progression.

### Final Answer:
Hence, the sequence \(9, 12, 16, 24, \ldots\) cannot form a G.P.

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