Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
a,2a,3a,4a

Answer

To determine whether the sequence \( a, 2a, 3a, 4a \) is an arithmetic progression (A.P.), we will follow these steps:

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

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

1. Difference between the second and first term:
\[ a_2 - a_1 = 2a - a = a \]

2. Difference between the third and second term:
\[ a_3 - a_2 = 3a - 2a = a \]

3. Difference between the fourth and third term:
\[ a_4 - a_3 = 4a - 3a = a \]

### Step 3: Check if the differences are equal
Since the differences between all consecutive terms are equal to \( a \):
- \( a_2 - a_1 = a \)
- \( a_3 - a_2 = a \)
- \( a_4 - a_3 = a \)

This confirms that the sequence \( a, 2a, 3a, 4a \) is indeed an A.P.

### Step 4: Find the common difference
The common difference \( d \) is:
\[ d = a \]

### Step 5: Write three more terms of the A.P.
To find the next three terms in the A.P., we will add the common difference \( d \) to the last term \( a_4 \):

1. Fifth term:
\[ a_5 = a_4 + d = 4a + a = 5a \]

2. Sixth term:
\[ a_6 = a_5 + d = 5a + a = 6a \]

3. Seventh term:
\[ a_7 = a_6 + d = 6a + a = 7a \]

### Final Result
The next three terms of the A.P. are \( 5a, 6a, 7a \).

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

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