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The pattern we spotted is that when you add 3 consecutive terms of the Fibonacci sequence, the total is double the third term.

First we tried 1+1+2=4, then we tried 2+3+5=10 and 5+8+13=26.

We noticed that the first and second term would add up to the third term so the total would be 2 multiplied by the 3rd term. So we decided to use algebra to prove this.

Of the three terms you are adding up, let the first term equal a and the second term equal b. The third term will be a+b.

The sum of your three consecutive terms will be a+b+a+b = 2a+2b = 2(a+b).

So, we have proven the total of three consecutive terms is 2 multiplied by the third term.