[VA] SRC#003- New Year 2019 Special

[Albert Chan]

Tried doing √3 with this matrix power method, noticed a pattern:
let M = {{1,3}, {1, 1}}

M^2 = {{3*1+1, 3*(1+1)}, {1+1, 3*1+1}} = {{4,6}, {2,4}}
M^3 = {{3*2+4, 3*(2+4)}, {2+4, 3*2+4}} = {{10,18}, {6,10}}

Right diagonal ratio stayed at 3.0, and left diagonal same numbers.
-> only need to do bottom row. Top row can be deduced if needed.
-> each matrix multiply required only 2 add, and 1 multiply

row 2 of M^4 = {6+10, 3*6+10} = {16, 28}
row 2 of M^5 = {16+28, 3*16+28} = {44, 76}
row 2 of M^6 = {44+76, 3*44+76} = {120, 208} ...

Doing the average of ratios, for M^6: √3 ~ ½(208/120 + 360/208) ~ ½(1.733333 + 1.730769) = 1.732051