Which recursive sequence would produce the sequence 1,0,-2,dots ? a_(1)=1 and a_(n)=3a_(n-1)-3 a_(1)=1 and a_(n)=-3a_(n-1)+3 a_(1)=1 and a_(n)=-2a_(n-1)+2 a_(1)=1 and a_(n)=2a_(n-1)-2

Question

Which recursive sequence would produce the sequence  1,0,-2,dots ?  a_(1)=1 and  a_(n)=3a_(n-1)-3  a_(1)=1 and  a_(n)=-3a_(n-1)+3  a_(1)=1 and  a_(n)=-2a_(n-1)+2  a_(1)=1 and  a_(n)=2a_(n-1)-2

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Given Sequence Determination: We are given the first term of the sequence, a_(1) = 1. We need to determine which recursive formula will generate the sequence 1, 0, -2, ... when applied starting from this first term.Testing First Option: Let's test the first option: a_(1) = 1 and a_(n) = 3a_(n-1) - 3. We will calculate the second term using the first term. a_(2) = 3a_(1) - 3 = 3*1 - 3 = 0. This matches the second term of the given sequence.Testing Second Option: Now, let's calculate the third term using the second term we just found. a_(3) = 3a_(2) - 3 = 3*0 - 3 = -3. However, the third term of the given sequence is -2, not -3. Therefore, this recursive formula does not produce the given sequence.Testing Third Option: Let's test the second option: a_(1) = 1 and a_(n) = -3a_(n-1) + 3. We will calculate the second term using the first term. a_(2) = -3a_(1) + 3 = -3*1 + 3 = 0. This matches the second term of the given sequence.Testing Fourth Option: Now, let's calculate the third term using the second term we just found. a_(3) = -3a_(2) + 3 = -3*0 + 3 = 3. However, the third term of the given sequence is -2, not 3. Therefore, this recursive formula also does not produce the given sequence.Let's test the third option: a_(1) = 1 and a_(n) = -2a_(n-1) + 2. We will calculate the second term using the first term. a_(2) = -2a_(1) + 2 = -2*1 + 2 = 0. This matches the second term of the given sequence.Now, let's calculate the third term using the second term we just found. a_(3) = -2a_(2) + 2 = -2*0 + 2 = 2. However, the third term of the given sequence is -2, not 2. Therefore, this recursive formula does not produce the given sequence.Finally, let's test the fourth option: a_(1) = 1 and a_(n) = 2a_(n-1) - 2. We will calculate the second term using the first term. a_(2) = 2a_(1) - 2 = 2*1 - 2 = 0. This matches the second term of the given sequence.Now, let's calculate the third term using the second term we just found. a_(3) = 2a_(2) - 2 = 2*0 - 2 = -2. This matches the third term of the given sequence. Therefore, this recursive formula produces the given sequence.

Which recursive sequence would produce the sequence $1,0,-2, \ldots$ ? $\newline$ $a_{1}=1$ and $a_{n}=3 a_{n-1}-3$ $\newline$ $a_{1}=1$ and $a_{n}=-3 a_{n-1}+3$ $\newline$ $a_{1}=1$ and $a_{n}=-2 a_{n-1}+2$ $\newline$ $a_{1}=1$ and $a_{n}=2 a_{n-1}-2$

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