Mathematics Test 43

Result

Mathematics Test 43

Score
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Rank
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TIME Taken - -

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Weekly Quiz Competition

Question 1
4 / -1

If X = {4ⁿ – 3n – 1 : n ∈ N} and Y = {9(n – 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to

A
N

B
Y – X

C
X

D
Y

Solution

Set X contains elements of the form
4ⁿ – 3n – 1 = (1 + 3)ⁿ – 3n – 1
= 3ⁿ + ⁿC_{n – 1} 3^{n – 1} . . . ^nC₂ 3²
= 9(3^{n – 2} + ⁿC_{n – 1} 3^{n – 1} . . . + ⁿC₂
Set X has natural numbers which are multiples of 9 (not all)
Set Y has all multiples of 9
X ∪ Y = Y
Hence, option D is correct.