If (S_{n+4}-S_{n-1}=485), what is the value of (n)?
Answer and explanation
Correct answer: 95
Here, \(S_k\) denotes the sum of the first \(k\) natural numbers. In \(S_{n+4}-S_{n-1}\), the terms from \(1\) to \(n-1\) cancel, leaving \(n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10\). Thus, \(5n+10=485\), so \(5n=475\) and \(n=95\). If \(n=96\), the difference would be \(490\), not \(485\). Exam tip: the sum of five consecutive terms equals \(5\) times their middle term, i.e. \(5(n+2)\).
Frequently asked questions
What is the correct answer to this question?
95
Why is this the correct answer?
Here, \(S_k\) denotes the sum of the first \(k\) natural numbers. In \(S_{n+4}-S_{n-1}\), the terms from \(1\) to \(n-1\) cancel, leaving \(n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10\). Thus, \(5n+10=485\), so \(5n=475\) and \(n=95\). If \(n=96\), the difference would be \(490\), not \(485\). Exam tip: the sum of five consecutive terms equals \(5\) times their middle term, i.e. \(5(n+2)\).
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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