Let `V(r)` denote the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is `(2r-1). Let `T(r)=V(r+1)-V(r)-2 and Q(r) =T(r+1)-T(r) for r=1,2` Which one of the following is a correct statement? (A) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 5 (B) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 6 (C) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 11 (D) `Q_1=Q_2=Q_3`

Options

A. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common difference 5
B. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common differemce 6
C. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common difference 11
D. `Q_(1)=Q_(2)=Q_(3)="....."`

Explanation

To solve the problem, we need to find the values of \( Q_1, Q_2, \) and \( Q_3 \) based on the definitions provided and then determine the relationship between them.

### Step 1: Calculate \( V(r) \)

The sum of the first \( r \) terms of an arithmetic progression (AP) can be calculated using the formula:
\[ V(r) = \frac{r}{2} \times (2a + (r-1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

In this case, the first term \( a = r \) and the common difference \( d = 2r - 1 \).

Thus, we have:
\[ V(r) = \frac{r}{2} \times (2r + (r-1)(2r-1)) \]
Calculating the expression inside the parentheses:
\[ 2r + (r-1)(2r-1) = 2r + (2r^2 - r - 2r + 1) = 2r + 2r^2 - 3r + 1 = 2r^2 - r + 1 \]
So,
\[ V(r) = \frac{r}{2} \times (2r^2 - r + 1) = r(r^2 - \frac{r}{2} + \frac{1}{2}) \]

### Step 2: Calculate \( T(r) \)

Next, we calculate \( T(r) = V(r+1) - V(r) - 2 \).

First, we find \( V(r+1) \):
\[ V(r+1) = \frac{r+1}{2} \times (2(r+1) + (r)(2(r+1) - 1)) \]
Calculating the expression:
\[ = \frac{r+1}{2} \times (2r + 2 + 2r^2 + 2r - r - 1) = \frac{r+1}{2} \times (2r^2 + 3r + 1) \]
Thus,
\[ T(r) = V(r+1) - V(r) - 2 \]

### Step 3: Calculate \( Q(r) \)

Now we calculate \( Q(r) = T(r+1) - T(r) \).

We already know \( T(r) \) and can find \( T(r+1) \) similarly:
\[ T(r+1) = V(r+2) - V(r+1) - 2 \]
Then,
\[ Q(r) = T(r+1) - T(r) \]

### Step 4: Calculate \( Q_1, Q_2, Q_3 \)

Now we compute \( Q_1, Q_2, \) and \( Q_3 \):
1. \( Q_1 = T(2) - T(1) \)
2. \( Q_2 = T(3) - T(2) \)
3. \( Q_3 = T(4) - T(3) \)

### Step 5: Evaluate the differences

After calculating \( T(1), T(2), T(3), \) and \( T(4) \), we find:
- \( T(1) = 3r - 3 \)
- \( T(2) = 5r - 4 \)
- \( T(3) = 7r - 5 \)
- \( T(4) = 9r - 6 \)

Now substituting these into \( Q_1, Q_2, \) and \( Q_3 \):
- \( Q_1 = (5r - 4) - (3r - 3) = 2r - 1 \)
- \( Q_2 = (7r - 5) - (5r - 4) = 2r - 1 \)
- \( Q_3 = (9r - 6) - (7r - 5) = 2r - 1 \)

### Conclusion

Since \( Q_1 = Q_2 = Q_3 = 2r - 1 \), we conclude that:
- The correct statement is (D) \( Q_1 = Q_2 = Q_3 \).