Which recursive sequence would produce the sequence 8,-28,80,dots ? a_(1)=8 and a_(n)=-4a_(n-1)-3 a_(1)=8 and a_(n)=-3a_(n-1)-4 a_(1)=8 and a_(n)=4a_(n-1)-4 a_(1)=8 and a_(n)=-4a_(n-1)+4

Question

Which recursive sequence would produce the sequence  8,-28,80,dots ?  a_(1)=8 and  a_(n)=-4a_(n-1)-3  a_(1)=8 and  a_(n)=-3a_(n-1)-4  a_(1)=8 and  a_(n)=4a_(n-1)-4  a_(1)=8 and  a_(n)=-4a_(n-1)+4

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Identify First Term: Identify the first term of the sequence. The first term given is a_(1) = 8.Calculate Second Term: Use the first recursive formula to calculate the second term. The first option is a_(n) = -4a_(n-1) - 3. Let's calculate a_(2): a_(2) = -4a_(1) - 3 = -4(8) - 3 = -32 - 3 = -35. This does not match the second term of the sequence, which is -28.Calculate Third Term: Use the second recursive formula to calculate the second term. The second option is a_(n) = -3a_(n-1) - 4. Let's calculate a_(2): a_(2) = -3a_(1) - 4 = -3(8) - 4 = -24 - 4 = -28. This matches the second term of the sequence.Verify Pattern - Third Formula: Use the second recursive formula to calculate the third term. Let's calculate a_(3) using the second option: a_(3) = -3a_(2) - 4 = -3(-28) - 4 = 84 - 4 = 80. This matches the third term of the sequence.Verify Pattern - Fourth Formula: Verify the pattern with the third recursive formula. The third option is a_(n) = 4a_(n-1) - 4. Let's calculate a_(2): a_(2) = 4a_(1) - 4 = 4(8) - 4 = 32 - 4 = 28. This does not match the second term of the sequence, which is -28.Calculate Third Term: Verify the pattern with the fourth recursive formula. The fourth option is a_(n) = -4a_(n-1) + 4. Let's calculate a_(2): a_(2) = -4a_(1) + 4 = -4(8) + 4 = -32 + 4 = -28. This matches the second term of the sequence, but we need to check the third term to be sure.Use the fourth recursive formula to calculate the third term. Let's calculate a_(3) using the fourth option: a_(3) = -4a_(2) + 4 = -4(-28) + 4 = 112 + 4 = 116. This does not match the third term of the sequence, which is 80.

Which recursive sequence would produce the sequence $8,-28,80, \ldots$ ? $\newline$ $a_{1}=8$ and $a_{n}=-4 a_{n-1}-3$ $\newline$ $a_{1}=8$ and $a_{n}=-3 a_{n-1}-4$ $\newline$ $a_{1}=8$ and $a_{n}=4 a_{n-1}-4$ $\newline$ $a_{1}=8$ and $a_{n}=-4 a_{n-1}+4$

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