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:
Can you prove the conjecture?
I wonder if there are any integers $n$ where $8n+1$ is a square number but $n$ is not a triangular number...
Can you prove that if $8n+1$ is a square number, $n$ must be a triangular number?
Can you use your theorem to devise a quick way to check whether the following numbers are triangular numbers?