For which value of \(n\) will \(S_n-S_{n-4}=282\), where \(S_n=1+2+\cdots+n\)?
Answer and explanation
Correct answer: 72
The governing concept is the difference between two sums of consecutive natural numbers. In \(S_n=1+2+\cdots+n\), all terms from 1 through \(n-4\) also occur in \(S_{n-4}\), so they cancel when the latter is subtracted. The remaining terms are \((n-3)+(n-2)+(n-1)+n\). Combining like terms gives \(4n-6\). Therefore, \(4n-6=282\), so \(4n=288\) and \(n=72\). Hence option C is correct. Checking the distractors confirms the result: for 70 the difference is 274, for 71 it is 278, and for 73 it is 286. Only 72 makes the difference exactly 282.
Frequently asked questions
What is the correct answer to this question?
72
Why is this the correct answer?
The governing concept is the difference between two sums of consecutive natural numbers. In \(S_n=1+2+\cdots+n\), all terms from 1 through \(n-4\) also occur in \(S_{n-4}\), so they cancel when the latter is subtracted. The remaining terms are \((n-3)+(n-2)+(n-1)+n\). Combining like terms gives \(4n-6\). Therefore, \(4n-6=282\), so \(4n=288\) and \(n=72\). Hence option C is correct. Checking the distractors confirms the result: for 70 the difference is 274, for 71 it is 278, and for 73 it is 286. Only 72 makes the difference exactly 282.
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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