Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`a,2a, 3a,4a,…`

Answer

To determine if the sequence \( a, 2a, 3a, 4a, \ldots \) forms an Arithmetic Progression (AP), we will follow these steps:

### Step 1: Identify the terms of the sequence
The given sequence is:
- First term \( a_1 = a \)
- Second term \( a_2 = 2a \)
- Third term \( a_3 = 3a \)
- Fourth term \( a_4 = 4a \)

### Step 2: Calculate the common difference
To check if the sequence is an AP, we need to find the common difference \( d \). The common difference is defined as the difference between consecutive terms.

1. Calculate \( d \) between the first and second terms:
\[ d = a_2 - a_1 = 2a - a = a \]

2. Calculate \( d \) between the second and third terms:
\[ d = a_3 - a_2 = 3a - 2a = a \]

3. Calculate \( d \) between the third and fourth terms:
\[ d = a_4 - a_3 = 4a - 3a = a \]

Since the common difference \( d \) is the same for all pairs of consecutive terms, we can conclude that the sequence is an AP.

### Step 3: Write three more terms of the AP
Now that we know the sequence is an AP with a common difference of \( d = a \), we can find the next three terms.

1. The fifth term \( a_5 \):
\[ a_5 = a_4 + d = 4a + a = 5a \]

2. The sixth term \( a_6 \):
\[ a_6 = a_5 + d = 5a + a = 6a \]

3. The seventh term \( a_7 \):
\[ a_7 = a_6 + d = 6a + a = 7a \]

### Final Result
The next three terms of the AP are:
- \( a_5 = 5a \)
- \( a_6 = 6a \)
- \( a_7 = 7a \)

### Summary
The sequence \( a, 2a, 3a, 4a, \ldots \) is an AP with a common difference \( d = a \). The next three terms are \( 5a, 6a, 7a \).

---