This is an old one but fun, and a good way to use algebra to show why a trick works. It’s a similar to showing how Magic Squares work. It’s not a formal proof as such, but I think it’s a good way to introduce the topic.
Once students have grasped the basic concept of a Fibonacci Series (something which, in my experience they often see at Primary School even if they can’t remember what it is called), then you are ready to start the trick.
Fibonacci series don’t have to start with a 1 and a 1 as in the diagram above. You start by asking students which two numbers they want to start with.
Then they get ready to be wowed with your powers of mental arithmetic. Tell them that you will be able to add up the first 10 digits of this sequence in your head faster than they can on calculators. Get one student up to the board to write down the numbers one by one. TOP TIP here: make sure you have the numbers 1-10 written in a vertical column and that the chosen student writes down each term in the sequence against the numbers. You should end up with something like this on your board:
As soon as term 7 goes up on the board, you start calculating. You should be able to find the sum of the first 10 terms before they even get to term 10 and this is why:
I quite like doing the calculation on a miniwhiteboard, then writing the answer face down on a students’ desk and then walking to the other side of the room. Once they have finally totalled the column of numbers on their calculator, you ask a student to have a look under the whiteboard.
And like all good magicians, you DO then go on to reveal the secrets of your trick!