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
\[
5m \sum_{r=m}^{2m} T_r \text{ is equal to:}
\]
\[ 5m \sum_{r=m}^{2m} T_r \text{ is equal to:} \]
Show Hint
- 112
- 142
- 126
- 98
The Correct Option is C
Solution and Explanation
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.
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