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Three Neighbours

Take three consecutive numbers and add them together. What do you notice?

Secondary Toughnuts

These secondary problems have not yet been solved. Can you be the first?

Three Consecutive Odd Numbers

How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?

Adding Odd Numbers

Age 11 to 16
Challenge Level

 

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?

The first four odd numbers are $1$, $3$, $5$ and $7$. The sum of these is $1+3+5+7=16$.


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?

The sum of the first $100$ odd numbers is $10,000$

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?


 

If you can find a proof which is different to the one in our proof sorter, then please do let us know by submitting it as a solution.

 

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).

 

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.