The Fibonacci sequence is F(0) = 1, F(1) = 1, F(n>1) = F(n-1) + F(n-2), giving 1 1 2 3 5 8 13 21 34 55 89... and a property I just noticed is that ΣₙF(n) for 0 ≤ n ≤ m is just F(m + 2) - 1.
We can observe that this property is true for m = 0: the sum is 1, m + 2 = 2, F(2) = 2, so F(2) - 1 = 1. When we advance the sum from m₀ to m₀ + 1, we are adding F(m₀ + 1) to it. F(m₀ + 3) = F(m₀ + 2) + F(m₀ + 1), by definition, so if this property is true for some m₀, it is also true for m₀ + 1. So by induction it is always true.
In some sense we should expect something like this to be true, since the Fibonacci sequence grows exponentially, but it was still a bit of a surprise to me to observe that 1+1+2+3+5+8+13+21+34 = 89-1.