The correct option is B 4n−3n−1
A) a(n)=8n+1=(9−1)n+1=9n−nC19n−1+..+(−1)n−1nCn−19+(−1)nnCn+1
⇒a(n)=9(9n−1−nC19n−2+..+(−1)n−1nCn−1)+(−1)nnCn+1=9λ+(−1)nnCn+1
if n is even, then a(n)=9λ+2 which is not divisible by 9
if n is odd, then a(n)=9λ which is divisible by 9
B) b(n)=4n−3n−1=(3+1)n−3n−1=3n+nC13n−1+...+nC232+nC13+nCn−3n−1
⇒b(n)=9(3n−2+nC13n−3+...+nC2)=9λ
Therefore, b(n) is divisible by 9.
C) c(n)=32n+3n+1=3(32n−1+n)+1=3λ+1
Therefore, c(n) is not divisible by 9.
D) d(n)=10n+1=(9+1)n+1=9n+nC19n−1+...+nCn+1
⇒d(n)=9(9n−1+nC19n−2+..+nCn−1)+2=9λ+2
Therefore, d(n) is not divisible by 9