Iff
Alison has been playing with numbers again. She started by choosing a triangular number, multiplied it by 8, and added 1. She noticed something interesting about her results...
Try a few examples. Can you make a conjecture?
Once you've made a conjecture of your own, click below to see what Alison noticed:
"If $T$ is a triangular number, $8T+1$ is a square number."
Can you prove the conjecture?
You might like to have a look at this Scrambled Proof and see if you can rearrange it into the original order.
Claire thought that she could use a picture to prove this conjecture. Can you use her picture to create another proof to show that the conjecture is true?
I wonder if there are any integers $k$ where $8k+1$ is a square number but $k$ is not a triangular number...
Can you prove that if $8k+1$ is a square number, $k$ must be a triangular number?
Here is a Scrambled Proof for this conjecture.
Can you use your theorem to devise a quick way to check whether the following numbers are triangular numbers?
- 6214
- 3655
- 7626
- 8656
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