a. B can be written as $1^k$(0+1)*
k=1 -> 1(0+1)*1(0+1)* y=(0+1)*1(0+1)*
k=2 -> 11(0+1)*1(0+1)*1(0+1)* y=(0+1)*1(0+1)*1(0+1)*
k=3 -> 111(0+1)*1(0+1)*1(0+1)*1(0+1)* y=(0+1)*1(0+1)*1(0+1)*1(0+1)*
B =1(0+1)*1(0+1)* $\bigcup$ 11(0+1)*1(0+1)*1(0+1)* $\bigcup$ 111(0+1)*1(0+1)*1(0+1)*1(0+1)* ..........
=(ε+11(0+1)*+111(0+1)*1(0+1)*+........)(1(0+1)*1(0+1)*)
$=(ε+11\sum^*)(1\sum^*1\sum^*)$
______________________________________________________________________
b.
k=1 -> 1(0*+0*10*) y=0*+0*10*
k=2 -> 11(0*+0*10*+0*10*10*)
k=3 -> 111(0*+0*10*+0*10*10*+0*10*10*10*)
C=1(0*+0*10*) $\bigcup$ 11(0*+0*10*+0*10*10*) $\bigcup$ 111(0*+0*10*+0*10*10*+0*10*10*10*)........
clearly there is no pattern
C is not regular