Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
1,3,9,27

Answer

To determine whether the sequence 1, 3, 9, 27 forms an Arithmetic Progression (A.P.), we will follow these steps:

### Step 1: Identify the terms of the sequence
The given sequence is:
- First term (a) = 1
- Second term (b) = 3
- Third term (c) = 9
- Fourth term (d) = 27

### Step 2: Calculate the differences between consecutive terms
To check if the sequence is an A.P., we need to find the differences between consecutive terms:

1. Difference between the second and first term:
\[ b - a = 3 - 1 = 2 \]

2. Difference between the third and second term:
\[ c - b = 9 - 3 = 6 \]

3. Difference between the fourth and third term:
\[ d - c = 27 - 9 = 18 \]

### Step 3: Compare the differences
Now we compare the differences we calculated:
- First difference: 2
- Second difference: 6
- Third difference: 18

Since the differences (2, 6, and 18) are not equal, we conclude that the sequence does not form an A.P.

### Conclusion
The sequence 1, 3, 9, 27 is **not an A.P.** Therefore, we cannot find a common difference, nor can we write three more terms of the sequence as an A.P.

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