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: you’re changing the question. There is a difference between
“sum of the first n positive odd integers” and
“sum of all positive integers up to n”
Make sure you know which is which.
Still don’t get it, can you give me an example?
What is the sum of the first n odd positive integers? It’s the first n terms of the sequence
1,3,5,7,9,...
What are you trying to do? You’re trying to find the sum of the first n odd positive integers up to n itself. If n = 5, you’ll only have 1,3,5 in the sequence. That isn’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 that’s what this question is all about.
(Note: why 2n - 1? Because that’s the $n$th term in the sequence)
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