解：因為$2a - 1$的算術(shù)平方根是$5，$所以$2a - 1 = 5^2 = 25，$解得$2a = 26，$$a = 13。$
因為$3a + b - 1$的算術(shù)平方根是$4，$所以$3a + b - 1 = 4^2 = 16。$
將$a = 13$代入$3a + b - 1 = 16，$得$3×13 + b - 1 = 16，$即$39 + b - 1 = 16，$$38 + b = 16，$解得$b = -22。$
所以$a + 2b = 13 + 2×(-22) = 13 - 44 = -31。$
答：$a + 2b$的值為$-31。$
