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Class 9MathematicsSequences and ProgressionsSum of first n natural numbersHardLevel 62

If \(S_n\) is the sum of the first \(n\) natural numbers, which of the following statements is always true for every positive integer \(n\)?

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Answer and explanation

Correct answer: \(8S_n+1\) is a perfect square

Using \(S_n=\frac{n(n+1)}{2}\), we get \(8S_n+1=4n(n+1)+1=(2n+1)^2\), a perfect square. However, \(S_n\) itself is not always odd or a square. Exam tip: substitute the standard sum formula to test such identities.

Tags

natural numberssum formulatriangular numbersperfect squaresequences and progressions

Frequently asked questions

What is the correct answer to this question?

\(8S_n+1\) is a perfect square

Why is this the correct answer?

Using \(S_n=\frac{n(n+1)}{2}\), we get \(8S_n+1=4n(n+1)+1=(2n+1)^2\), a perfect square. However, \(S_n\) itself is not always odd or a square. Exam tip: substitute the standard sum formula to test such identities.

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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