Which recursive sequence would produce the sequence 2,3,6,dots ? a_(1)=2 and a_(n)=3a_(n-1)-3 a_(1)=2 and a_(n)=-5a_(n-1)+4 a_(1)=2 and a_(n)=-3a_(n-1)+3 a_(1)=2 and a_(n)=4a_(n-1)-5

Question

Which recursive sequence would produce the sequence  2,3,6,dots ?  a_(1)=2 and  a_(n)=3a_(n-1)-3  a_(1)=2 and  a_(n)=-5a_(n-1)+4  a_(1)=2 and  a_(n)=-3a_(n-1)+3  a_(1)=2 and  a_(n)=4a_(n-1)-5

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Test Formula 1: Let's test each recursive formula by applying it to the given initial value and checking if it produces the sequence 2, 3, 6, ... .  We start with the first option: a_(1)=2 and  a_(n)=3a_(n-1)-3  We know a_(1)=2, so let's find a_(2): a_(2)=3a_(1)-3 a_(2)=3*2-3 a_(2)=6-3 a_(2)=3  Now let's find a_(3): a_(3)=3a_(2)-3 a_(3)=3*3-3 a_(3)=9-3 a_(3)=6  So far, the sequence matches: 2, 3, 6. Let's check one more term to be sure. a_(4)=3a_(3)-3 a_(4)=3*6-3 a_(4)=18-3 a_(4)=15  The sequence we have from this formula is 2, 3, 6, 15, which does not match the expected sequence of 2, 3, 6, ... because the next term should be a multiple of the previous term, not 15.Test Formula 2: Let's test the second option: a_(1)=2 and  a_(n)=-5a_(n-1)+4  We know a_(1)=2, so let's find a_(2): a_(2)=-5a_(1)+4 a_(2)=-5*2+4 a_(2)=-10+4 a_(2)=-6  This does not match the second term of the sequence, which is 3, so this option is incorrect.Test Formula 3: Let's test the third option: a_(1)=2 and  a_(n)=-3a_(n-1)+3  We know a_(1)=2, so let's find a_(2): a_(2)=-3a_(1)+3 a_(2)=-3*2+3 a_(2)=-6+3 a_(2)=-3  This does not match the second term of the sequence, which is 3, so this option is incorrect.Test Formula 4: Finally, let's test the fourth option: a_(1)=2 and  a_(n)=4a_(n-1)-5  We know a_(1)=2, so let's find a_(2): a_(2)=4a_(1)-5 a_(2)=4*2-5 a_(2)=8-5 a_(2)=3  Now let's find a_(3): a_(3)=4a_(2)-5 a_(3)=4*3-5 a_(3)=12-5 a_(3)=7  This does not match the third term of the sequence, which is 6, so this option is incorrect.

Which recursive sequence would produce the sequence $2,3,6, \ldots$ ? $\newline$ $a_{1}=2$ and $a_{n}=3 a_{n-1}-3$ $\newline$ $a_{1}=2$ and $a_{n}=-5 a_{n-1}+4$ $\newline$ $a_{1}=2$ and $a_{n}=-3 a_{n-1}+3$ $\newline$ $a_{1}=2$ and $a_{n}=4 a_{n-1}-5$

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