If a positive integer \(S\) can be written as the sum of the first \(n\) natural numbers, which of the following quantities must be an odd perfect square?
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Answer and explanation
Correct answer: \(8S+1\)
The sum is \(S=\frac{n(n+1)}{2}\). Hence \(8S+1=4n(n+1)+1=(2n+1)^2\), which is an odd perfect square. Exam tip: use this identity to test whether a number is triangular.
Frequently asked questions
What is the correct answer to this question?
\(8S+1\)
Why is this the correct answer?
The sum is \(S=\frac{n(n+1)}{2}\). Hence \(8S+1=4n(n+1)+1=(2n+1)^2\), which is an odd perfect square. Exam tip: use this identity to test whether a number is triangular.
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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