First, set this equal to x:
x=11+12+11+12+...
We can replace the bottom part with x, shown below:
x=11+12+x
Now, solve for x:
x=12+x+12+xx=13+x2+xx=2+x3+xx2+3x=2+xx2+2x−2=0
After using the quadratic formula or completing the square, we find that x=−1−√3 or x=√3−1. Since x must be positive, the negative solution is extraneous, and therefore the answer is √3−1
.