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Question

if A=4n-3n-1:nbelongstoN and B=9(n-1):nbelongstoN,then


A

BA

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B

AB=N

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C

AB

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D

None of these

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Solution

The correct option is D

None of these


The explanation for the correct option:

Step1. The given data:

Given sets

A=4n-3n-1:nbelongstoN

and B=9(n-1):nbelongstoN

Set A can be expressed as

A=(3+1)n3n1=C0n30+C1n31+C2n32+.+Cn-1n3n-1+Cnn3n3n1=1+3n+C2n32+...+Cnn3n3n1=9(C2n+C3n(3)+....)

So, 4n3n-1,is a multiple of 9for n2

Now, Put n=1, in 4n3n-1, we get

4n3n-1=431=0

Now, For n=2, in 4n3n-1, we get

4n3n-1=1661=9

Hence, 4n3n1is a multiple of 9 for all nN

Step2. Find the relation on the given sets:

And the set B=9(n-1):nbelongstoN

Now, Put n=1, in 9(n-1), we get

9(n-1)=9(1-1)=0

Now, Put n=2 in 9(n-1), we get

9(n-1)=9(2-1)=9

Hence, 4n3n1is a multiple of 9 for all nN

AB and BA

A=B

Hence, Option ‘D’ is Correct.


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