Why do I get the Fibonacci sequence when I start with 1 and keep using the + sign?

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I played with this on a calculator and when I entered 1 and kept hitting the + button, I got the noticeable Fibonacci sequence! Can someone explain to me why this happens?

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$$1+1=2\\ 1+2=3\\ 2+3=5\\ 3+5=8$$

So your calculator is adding the current number to the previous term of the Fibonacci sequence.