If the sum of the first (n) natural numbers is (S_n), what is (S_{n+1}-S_{n-1}) equal to?
Answer and explanation
Correct answer: \(2n+1\)
\(S_{n+1}=1+2+\cdots+n+(n+1)\) and \(S_{n-1}=1+2+\cdots+(n-1)\). On subtracting, the common terms cancel, leaving \(n+(n+1)=2n+1\). Hence, the correct answer is \(2n+1\). \(n+1\) is only the newly added last term; the term \(n\) must also be included. Exam tip: Expand the two sums and identify the terms that cancel.
Frequently asked questions
What is the correct answer to this question?
\(2n+1\)
Why is this the correct answer?
\(S_{n+1}=1+2+\cdots+n+(n+1)\) and \(S_{n-1}=1+2+\cdots+(n-1)\). On subtracting, the common terms cancel, leaving \(n+(n+1)=2n+1\). Hence, the correct answer is \(2n+1\). \(n+1\) is only the newly added last term; the term \(n\) must also be included. Exam tip: Expand the two sums and identify the terms that cancel.
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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