What is the value of \(S_{100}-2S_{50}\), where \(S_n=1+2+\cdots+n\)?
Answer and explanation
Correct answer: 2500
Use the standard formula for the sum of the first \(n\) natural numbers: \(S_n=\frac{n(n+1)}2\). Thus \(S_{100}=\frac{100\cdot101}{2}=5050\), and \(S_{50}=\frac{50\cdot51}{2}=1275\). The expression contains twice \(S_{50}\), so \(2S_{50}=2\cdot1275=2550\). Therefore, \(S_{100}-2S_{50}=5050-2550=2500\), which is option C. A common error is to subtract \(S_{50}\) only once, giving 3775, or to mishandle the factor 2. Options A, B, and D do not satisfy the direct calculation.
Frequently asked questions
What is the correct answer to this question?
2500
Why is this the correct answer?
Use the standard formula for the sum of the first \(n\) natural numbers: \(S_n=\frac{n(n+1)}2\). Thus \(S_{100}=\frac{100\cdot101}{2}=5050\), and \(S_{50}=\frac{50\cdot51}{2}=1275\). The expression contains twice \(S_{50}\), so \(2S_{50}=2\cdot1275=2550\). Therefore, \(S_{100}-2S_{50}=5050-2550=2500\), which is option C. A common error is to subtract \(S_{50}\) only once, giving 3775, or to mishandle the factor 2. Options A, B, and D do not satisfy the direct calculation.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Sum of first n natural numbers.
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