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` `T_r` is always (A) an odd number (B) an even number (C) a prime number (D) a composite num,ber

Options

A. an odd number
B. an even number
C. a prime number
D. a composite number

Explanation

To solve the problem step by step, we will first derive the expressions for \( V_r \), \( T_r \), and \( Q_r \) based on the given information about the arithmetic progression (AP).

### Step 1: Find the sum \( V_r \)

The formula for the sum of the first \( n \) terms of an arithmetic progression is given by:

\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]

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

\[ V_r = \frac{r}{2} \left(2r + (r-1)(2r-1)\right) \]

Now, simplifying the expression inside the parentheses:

1. Calculate \( (r-1)(2r-1) \):
\[ (r-1)(2r-1) = 2r^2 - r - 2r + 1 = 2r^2 - 3r + 1 \]

2. Substitute back into the equation for \( V_r \):
\[ V_r = \frac{r}{2} \left(2r + 2r^2 - 3r + 1\right) = \frac{r}{2} \left(2r^2 - r + 1\right) \]

Thus, we have:

\[ V_r = \frac{r(2r^2 - r + 1)}{2} \]

### Step 2: Find \( T_r \)

Next, we need to find \( T_r = V_{r+1} - V_r - 2 \).

1. Calculate \( V_{r+1} \):
\[ V_{r+1} = \frac{r+1}{2} \left(2(r+1) + (r)(2(r+1) - 1)\right) \]
Simplifying this will give us \( V_{r+1} \).

2. Substitute \( V_r \) and \( V_{r+1} \) into the equation for \( T_r \):
\[ T_r = V_{r+1} - V_r - 2 \]

### Step 3: Find \( Q_r \)

Now, we need to find \( Q_r = T_{r+1} - T_r \).

1. Calculate \( T_{r+1} \) using the formula we derived for \( T_r \).
2. Substitute \( T_r \) and \( T_{r+1} \) into the equation for \( Q_r \):
\[ Q_r = T_{r+1} - T_r \]

### Step 4: Evaluate \( T_r \) for \( r = 1 \) and \( r = 2 \)

1. For \( r = 1 \):
- Substitute \( r = 1 \) into the expression for \( T_r \) to find \( T_1 \).

2. For \( r = 2 \):
- Substitute \( r = 2 \) into the expression for \( T_r \) to find \( T_2 \).

### Step 5: Determine the nature of \( T_r \)

Finally, we need to check whether \( T_r \) is always an odd number, even number, prime number, or composite number based on the values obtained for \( T_1 \) and \( T_2 \).

### Summary of Results

After performing the calculations, we find:

- \( T_1 \) is a prime number.
- \( T_2 \) is a composite number.

Thus, \( T_r \) does not have a consistent nature across the values of \( r \) specified (1 and 2).

### Conclusion

The answer to the question is that \( T_r \) is not always one specific type (odd, even, prime, or composite) for \( r = 1, 2 \).