Question
If X = {4n - 3n - 1 : n ∈ N} and Y = { 9(n-1) : n ∈ N}, then X ∪ Y is equal to
Answer: Option B
:
B
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,nCn−1=nC1.....soon.)=9[nC2+nC3(3)+......+nC43n−1]
∴4n−3n−1is a multiple of 9 for n≥2.
Forn=1,4n−3n−1=4−3−1=0Forn=2,4n−3n−1=16−6−1=9∴4n−3n−1 is a multiple of 9 for all nϵN
∴X contains elements, which are multiples of 9, and clearly Y contains all multiples of 9.
∴X⊂Yi.e.,X∪Y=Y
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:
B
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,nCn−1=nC1.....soon.)=9[nC2+nC3(3)+......+nC43n−1]
∴4n−3n−1is a multiple of 9 for n≥2.
Forn=1,4n−3n−1=4−3−1=0Forn=2,4n−3n−1=16−6−1=9∴4n−3n−1 is a multiple of 9 for all nϵN
∴X contains elements, which are multiples of 9, and clearly Y contains all multiples of 9.
∴X⊂Yi.e.,X∪Y=Y
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