Math 2.0

Combinatorics Theorem

Sum of the first n odd positive integers

Builds on Mathematical induction

Statement

Let nn be a positive integer. Then the nn smallest odd positive integers add up to the square of nn: 1+3+5+⋯+(2n−1)=∑i=1n(2i−1)=n2.1 + 3 + 5 + \dots + (2n - 1) = \sum_{i=1}^{n} (2i - 1) = n^2.

Remarks

For n=1,2,3,4n = 1, 2, 3, 4 the sums are 1,4,9,161, 4, 9, 16. A proof by induction uses n2+(2n+1)=(n+1)2n^2 + (2n + 1) = (n + 1)^2.

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Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_induction_proofs.ptx#L104, source/s_induction_proofs.ptx:104
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Tag9PESCNHE7Q
URIm2://9PESCNHE7Q
Kindstatement, theorem
WritingW1 stated
MSC05A19
StatusThis copy of the site has no build data. Run m2 status 9PESCNHE7Q.
UsesMathematical induction
Changed4 November 2026

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