Show that the sum of the first n positive odd integers,1 +3 +5 + ...

Question

Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

Accepted Answer

Hey everyone here we are asked to show that the sum of the given series is two n squared minus n. Now, before we can begin solving this problem, we first need to recall the formula for the sum of an arithmetic series which is s the sum is equal to and divided by two times the quantity of a sub one plus a sub N. So now we want to identify each of these terms here aside from N. Because it's included in our solution, so we can begin with a sub one. A sub one is our first term and we are given the first term in the series which is one, so a sub one is just one. And now moving on to a sub end, our end term is this expression given at the end of the series which is four n minus three. And lastly we are told to show these psalms. So we're told our variable S. Here, our variable S. Is the sum to n squared minus n. And so now since we have identified each of these terms, we can just plug it into our formula. So beginning with the left hand side with S we again have to an squared minus n. Then we have is equal to and divided by two times the quantity of a sub one which is one plus a sub N, which is again four n minus three. And now all we need to do is simplify the right hand side and see if it matches up with the left hand side. So again on the left hand side we have two N squared minus N and then we have is equal to and for the right hand side we can just factor in this end divided by two. In doing this, we have N divided by two plus four and squared divided by two minus three, N, divided by two. And now all we have to do is add these terms together and simplify. And we find that the left hand side is once again to n squared minus n. And now our simplified right hand side is also to n squared minus n. And so since both sides of the equation are the same, we see that the solution of our given arithmetic series is indeed to end squared minus n. And with this we are left with a true statement and answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

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