If X=4n-3n-1; n∈N and Y=9n-1:n∈N, then X∪Y is equal to

Since, 4n-3n-1=(3+1)n-3n-1=3n+nC13n-1+nC23n-2+ ...+nCn-13+nCn-3n-1=nC232+nC3.33+...+nCn3n, (nC0=nCn, nC1=nCn-1etc.)=9nC2+nC3(3)+....+nCn3n-1∴4n-3n-1 is a multiple of 9 for n≥2.For n=1, 4n-3n-1=4-3-1=0,For n=2, 4n-3n-1=16-6-1=9∴4n-3n-1 is a multiple of 9 for all n∈N∴ X contains elements, which are multiples of 9, and clearly Y contains all multiples of 9.∴X⊆Y i.e.X∪Y=Y