If \(S_n-S_{n-3}=114\), what is the value of \(n\)?
Answer and explanation
Correct answer: 39
Since \(S_n=1+2+\cdots+n\), subtracting \(S_{n-3}=1+2+\cdots+(n-3)\) leaves the last three terms: \(S_n-S_{n-3}=(n-2)+(n-1)+n=3n-3\). Set this equal to 114: \(3n-3=114\), so \(3n=117\) and \(n=39\). Therefore option C is correct. The original option list incorrectly placed 40 as option C, so it has been corrected to 39 to ensure that the question has one valid answer. Options 36, 38, and 42 give differences 105, 111, and 123 respectively, not 114.
Frequently asked questions
What is the correct answer to this question?
39
Why is this the correct answer?
Since \(S_n=1+2+\cdots+n\), subtracting \(S_{n-3}=1+2+\cdots+(n-3)\) leaves the last three terms: \(S_n-S_{n-3}=(n-2)+(n-1)+n=3n-3\). Set this equal to 114: \(3n-3=114\), so \(3n=117\) and \(n=39\). Therefore option C is correct. The original option list incorrectly placed 40 as option C, so it has been corrected to 39 to ensure that the question has one valid answer. Options 36, 38, and 42 give differences 105, 111, and 123 respectively, not 114.
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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