Tree7

The pattern is $2^k-1$ where $k$ is the position $\therefore$ the $7th$ figure has $\boxed{2^7-1=127}$ branches

Another way
2^0+2^1+2^2+2^3+2^4+2^5+2^6=127 :p

s an exponential growth. Let f be an exponential function describing how many line segments there are in the xth picture. f(x) = 2^x -1 f(7) = 128-1 = 127

If we look at the image we can see that the pattern is : 1,3,7,15,31,63,?. Now: 1=0+2^0 3=1+2^1 7=3+2^2 (or) the nth term is 2^n -1 so the 7th term is 2^7 - 1

We obtain the following sequence by the given gif (2 -1), (2^2 -1), (2^3 -1),(2^3 -1), (2^4 -1),.......(2^7 -1). Hence the required line segments are = 2^7 - 1 = 127

Long and unnecessary explanations:

For all 12 fractals, the number will always end in 1,3,7,5,1,3,7,5,1,3,7,5 in that order, and the front number will follow the pattern of n 2, n 2, n 2+1, n 2+1, n 2, n 2 so on.. and will have a result as follow 0,0,0,1,3,6,12,25,.. so we just have to pair those front and end numbers.

0-1, 0-3, 0-7, 1-5, 3-1, 6-3, 12-7, 25-5, 51-1, 102-3, 204-7, 409-5, and so on.

For the 13th towards 20th are:

819-1, 1638-3, 3276-7, 6553-5, 13107-1, 26214-3, 52428-7 and 104857-5.