This problem follows on from Adding Odd Numbers.
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.
One method of proof is called Proof by Induction, which can be helpful when trying to prove something linked to an integer, $n$. Click the button below for a brief explanation of Proof By Induction.
These two steps are enough to show that the proposition is true for all integer values $n$. If we can show that $P(1)$ is true, and that $P(k)$ true $\implies P(k+1)$ is true, then we have:
Below is a proof that has also been scrambled up.
Can you rearrange it into its original order?
Read the article Proof by Induction for more detailed explanations of how this works, and some more examples.
To see a different proof to the one above, take a look at Adding Odd Numbers.
You might like to try to use Proof by Induction to prove the proposition in OK! Now prove it.