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

Given an arithmetic progression (A.P.) and specific terms, calculate \( 5m \sum_{r=m}^{2m} T_r \).
Step 1: Determine the common difference \( d \) using the provided terms \( T_m = \frac{1}{25} \) and \( T_{25} = \frac{1}{20} \). 
Step 2: Express the \( r^{th} \) term, \( T_r \), as a function of \( d \) using the general A.P. term formula. 
Step 3: Compute the sum \( \sum_{r=m}^{2m} T_r \) utilizing the A.P. sum formula. 
Step 4: Multiply the sum by \( 5m \) to obtain the final result. 

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