So the wording of the following question is really messing with me. The sum of the first n positive odd integers should be up to n not n^2 no? So the comparison would be between the sum of the first positive odd integers up to n, and n^2

Watch out: youâ€™re changing the question. There is a difference between

â€śsum of the first *n* positive odd integersâ€ť and

â€śsum of all positive integers up to *n*â€ť

Make sure you know which is which.

Still donâ€™t get it, can you give me an example?

What is the sum of the first n odd positive integers? Itâ€™s the first n terms of the sequence

1,3,5,7,9,...

What are you trying to do? Youâ€™re trying to find the sum of the first n odd positive integers up to n itself. If n = 5, youâ€™ll only have 1,3,5 in the sequence. That isnâ€™t the same as finding the first n terms, which includes 7 and 9 as well.

Now, what **is** true is that

1 + 3 + 5 + 7 + 9 + ... + (2n - 1) = n^2

and thatâ€™s what this question is all about.

(Note: why 2n - 1? Because thatâ€™s the $n$th term in the sequence)

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