Verify that each of the following is an A.P., and then write its next three terms.
`a + b , (a +1)+ b, (a +1 ) + (b +1),...`

Answer

To verify that the sequence \( a + b, (a + 1) + b, (a + 1) + (b + 1), \ldots \) is an arithmetic progression (A.P.) and to find its next three terms, we can follow these steps:

### Step 1: Identify the terms
The given sequence consists of the following terms:
1. First term (\(T_1\)): \(a + b\)
2. Second term (\(T_2\)): \((a + 1) + b = a + b + 1\)
3. Third term (\(T_3\)): \((a + 1) + (b + 1) = a + b + 2\)

### Step 2: Check the common difference
To verify if the sequence is an A.P., we need to check if the difference between consecutive terms is constant.

- Calculate the difference between the second and first terms:
\[ T_2 - T_1 = (a + b + 1) - (a + b) = 1 \]

- Calculate the difference between the third and second terms:
\[ T_3 - T_2 = (a + b + 2) - (a + b + 1) = 1 \]

Since both differences are equal (1), we can conclude that the sequence is indeed an A.P.

### Step 3: Write the next three terms
In an A.P., the next terms can be found by adding the common difference to the last term.

- Fourth term (\(T_4\)):
\[ T_4 = T_3 + (T_2 - T_1) = (a + b + 2) + 1 = a + b + 3 \]

- Fifth term (\(T_5\)):
\[ T_5 = T_4 + (T_2 - T_1) = (a + b + 3) + 1 = a + b + 4 \]

- Sixth term (\(T_6\)):
\[ T_6 = T_5 + (T_2 - T_1) = (a + b + 4) + 1 = a + b + 5 \]

### Final Result
The next three terms of the sequence are:
1. \(T_4 = a + b + 3\)
2. \(T_5 = a + b + 4\)
3. \(T_6 = a + b + 5\)