Question:
If
\(\dfrac{48}{2.3}+\dfrac{47}{3.4}+\dfrac{46}{4.5}+\cdots+\dfrac{2}{48.49}+\dfrac{1}{49.50}\)
\(=\dfrac{51}{2}+\mathrm{K}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{50}\right)\), then
\(\mathrm{K}\) equals :
If
\(\dfrac{48}{2.3}+\dfrac{47}{3.4}+\dfrac{46}{4.5}+\cdots+\dfrac{2}{48.49}+\dfrac{1}{49.50}\)
\(=\dfrac{51}{2}+\mathrm{K}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{50}\right)\), then
\(\mathrm{K}\) equals :
\(\dfrac{48}{2.3}+\dfrac{47}{3.4}+\dfrac{46}{4.5}+\cdots+\dfrac{2}{48.49}+\dfrac{1}{49.50}\)
\(=\dfrac{51}{2}+\mathrm{K}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{50}\right)\), then
\(\mathrm{K}\) equals :
Show Hint
The general term is \(\dfrac{50-m}{m(m+1)}\); split it into \(\dfrac{50}{m(m+1)}-\dfrac1{m+1}\) and sum.
Updated On: Nov 12, 2026
- \(-1\)
- \(-\dfrac{1}{2}\)
- 1
- 2
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The Correct Option is A
Solution and Explanation
The terms are \(\dfrac{50-m}{m(m+1)}\) for \(m=2,3,\dots,49\) (m = 2 gives 48/(2.3), m = 49 gives 1/(49.50)).
Write \(\dfrac{50-m}{m(m+1)}=\dfrac{50}{m(m+1)}-\dfrac{m}{m(m+1)}=50\left(\dfrac1m-\dfrac1{m+1}\right)-\dfrac1{m+1}\).
Sum of the first part (telescoping): \(50\left(\dfrac12-\dfrac1{50}\right)=25-1=24\).
Sum of the second part: \(\sum_{m=2}^{49}\dfrac1{m+1}=\dfrac13+\dfrac14+\dots+\dfrac1{50}=H-\dfrac32\), where \(H=1+\dfrac12+\dots+\dfrac1{50}\).
Total \(=24-\left(H-\dfrac32\right)=\dfrac{51}{2}-H\).
Comparing with \(\dfrac{51}{2}+\mathrm{K}H\) gives \(\mathrm{K}=-1\).
Answer: (1) \(-1\)
Write \(\dfrac{50-m}{m(m+1)}=\dfrac{50}{m(m+1)}-\dfrac{m}{m(m+1)}=50\left(\dfrac1m-\dfrac1{m+1}\right)-\dfrac1{m+1}\).
Sum of the first part (telescoping): \(50\left(\dfrac12-\dfrac1{50}\right)=25-1=24\).
Sum of the second part: \(\sum_{m=2}^{49}\dfrac1{m+1}=\dfrac13+\dfrac14+\dots+\dfrac1{50}=H-\dfrac32\), where \(H=1+\dfrac12+\dots+\dfrac1{50}\).
Total \(=24-\left(H-\dfrac32\right)=\dfrac{51}{2}-H\).
Comparing with \(\dfrac{51}{2}+\mathrm{K}H\) gives \(\mathrm{K}=-1\).
Answer: (1) \(-1\)
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