Exploring numbers, thinking like a mathematician and proof

I had a conversation today with a non-mathematician friend who said that he recently discovered an amazing fact about numbers.

He discovered that the sum of all the powers of 2 up to a certain point is equal to the next power of 2 minus 2. As an equation:

I’ll be considering this for n > 1 since 2¹=2 and I’m interesting in summing up powers of 2.

Just to verify this in your mind, it’s nice to try an example.

When n = 3 say;

The great thing about this conversation was that he was so enthusiastic about the fact that he had discovered this and also amazed by the rule itself. When you think about it, it is pretty spectacular that this rule holds for all powers of 2. It made me wonder if I give students the chance to explore numbers enough – come to think of it, I don’t think I do. Thinking like any mathematician, it wasn’t good enough for me that this rule held for a few of the simple cases. It had to be proved. I didn’t get a chance to ask my friend whether he felt uneasy about the fact that this rule could break down for some large power of 2 that he hadn’t tried yet. I wondered whether I have to see this proved because I was trained to do this at university or whether I am the sort of person that requires proof and I would have been sceptical even without university training? Maybe it’s a bit of both.

Anyway, here’s a proof that I devised based on a simple trick and also a proof by induction. Does anyone have another way of proving this?


Proof by Induction

Oh and one more thing…I still can’t believe that it took me a full day to realise that this could easily be verified using the Geometric Series Formula. The right hand side of the equation is a geometric series with first term 2 and common ratio 2 – how did I miss that one???

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