Adding odd numbers
Problem
The sum of the first three odd numbers is $1+3+5 = 9$
What are the first four odd numbers? What is the sum of the first four odd numbers?
What is the sum of the first five odd numbers?
What is the sum of the first two odd numbers?
What if we only have the first odd number in the sum?
What is the sum of the first six, or ten odd numbers?
Do you notice anything special about your results?
Can you predict what the sum of the first $100$ odd numbers will be?
Can you predict what the sum of the first $n$ odd numbers will be?
Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true, and look to proofs to provide rigorous and convincing arguments and justifications.
Can you prove that the sum of the first $n$ odd numbers is $n^2$?
Below is a proof that has been scrambled up.
Can you rearrange it into its original order?
Click on student solutions to see some different proofs that students submitted.
Extension:
Can you show diagrammatically that the sum of the first $n$ odd numbers is $n^2$?
You can try using Proof by Induction to prove the same result in the problem Adding Odd Numbers (part 2).
Student Solutions
Shubhangee from Buckler's Mead Academy in England and Sunhari from British School Muscat sent diagrammatic proofs. This is Shubhangee's proof:
Moncef from London Academy in Morocco constructed a similar but slightly different proof to the one in the proof sorter. Click here to see Moncef's proof.
Sunhari also submitted a proof by induction:
Let P($n$) be the statement 'the sum of the first $n$ odd numbers is equal to $n^2$.
P(1): 1 = 1^2
P(1) is true.
Suppose it is true for P($k$),
$1+3+… +(2k-1) = k^2$
Then, P($k+1$)
$1+3+…+[2(k+1) -1]
\= k^2 + [2(k+1) -1]
\= k^2 + 2k +2 -1
\= k^2 + 2k + 1
\= (k+1)^2$
Therefore whenever P($k$) is true, P($k+1$) is also true. So since P(1) is true, P(2) must also be true, and so P(3) must also be true, and so on. This means P($n$) must be true for all positive integers $n$.
Teachers' Resources
Why do this problem?
This problem, along with the rest of the problems in the Proof for All (st)ages feature, provides an excellent context for observing, conjecturing and thinking about proof, and for appreciating the power of algebra.
Possible approach
These printable cards for sorting may be useful:
Adding Odd Numbers Proof Sort
Key question
Is there a way to represent the sum of odd numbers that will help to explain the patterns you noticed?
Possible support
Encourage students to work in pairs on the proof sorting exercise.