Question

Let \( T_r \) be the \( r^{th} \) term of an A.P. If for some \( m \), \( T_m = \frac{1}{25} \), \( T_{25} = \frac{1}{20} \), and \( \sum_{r=1}^{25} T_r = 13 \), then

\[ 5m \sum_{r=m}^{2m} T_r \text{ is equal to:} \]

• A. 112
• B. 142
• C. 126
• D. 98

Hint:

In an A.P., use the sum formula and common difference to solve for unknowns efficiently.

Answer 1

The Correct Option is C

We are given the terms of an A.P. and need to find the value of \( 5m \sum_{r=m}^{2m} T_r \). 
Step 1: Use the given values \( T_m = \frac{1}{25} \) and \( T_{25} = \frac{1}{20} \) to solve for the common difference \( d \). 
Step 2: Use the general formula for the \( r^{th} \) term of an A.P. to express \( T_r \) in terms of \( d \). 
Step 3: Use the sum formula for an A.P. to find \( \sum_{r=m}^{2m} T_r \). 
Step 4: Multiply the result by \( 5m \) to compute the final value. 

Final Conclusion: The value of \( 5m \sum_{r=m}^{2m} T_r \) is 126, which is Option 3.