If (S_{n+4}-S_n=490), what is the value of (n)?
Answer and explanation
Correct answer: 120
Here, \(S_n\) denotes the sum of the first \(n\) natural numbers. Thus, \(S_{n+4}-S_n\) leaves only the next four terms: \((n+1)+(n+2)+(n+3)+(n+4)=4n+10\). Therefore, \(4n+10=490\), so \(4n=480\) and \(n=120\). For the closest distractor, \(n=121\) would give a difference of \(494\), not 490. Exam tip: Write \(S_{n+k}-S_n\) as the sum of terms from \(n+1\) to \(n+k\).
Frequently asked questions
What is the correct answer to this question?
120
Why is this the correct answer?
Here, \(S_n\) denotes the sum of the first \(n\) natural numbers. Thus, \(S_{n+4}-S_n\) leaves only the next four terms: \((n+1)+(n+2)+(n+3)+(n+4)=4n+10\). Therefore, \(4n+10=490\), so \(4n=480\) and \(n=120\). For the closest distractor, \(n=121\) would give a difference of \(494\), not 490. Exam tip: Write \(S_{n+k}-S_n\) as the sum of terms from \(n+1\) to \(n+k\).
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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