Which recursive sequence would produce the sequence 5,-16,68,dots ? a_(1)=5 and a_(n)=-3a_(n-1)-1 a_(1)=5 and a_(n)=-4a_(n-1)+4 a_(1)=5 and a_(n)=-a_(n-1)-3 a_(1)=5 and a_(n)=4a_(n-1)-4

Question

Which recursive sequence would produce the sequence  5,-16,68,dots ?  a_(1)=5 and  a_(n)=-3a_(n-1)-1  a_(1)=5 and  a_(n)=-4a_(n-1)+4  a_(1)=5 and  a_(n)=-a_(n-1)-3  a_(1)=5 and  a_(n)=4a_(n-1)-4

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Test Formula -1: Let's test each given recursive sequence formula by applying it to the initial term a_(1)=5 and see which one produces the sequence 5, -16, 68, ... .  First, we test the recursive sequence formula a_(n)=-3a_(n-1)-1 with a_(1)=5.  a_(2) = -3 * a_(1) - 1 = -3 * 5 - 1 = -15 - 1 = -16 a_(3) = -3 * a_(2) - 1 = -3 * (-16) - 1 = 48 - 1 = 47  The third term does not match the given sequence (68), so this formula is incorrect.Test Formula -2: Next, we test the recursive sequence formula a_(n)=-4a_(n-1)+4 with a_(1)=5.  a_(2) = -4 * a_(1) + 4 = -4 * 5 + 4 = -20 + 4 = -16 a_(3) = -4 * a_(2) + 4 = -4 * (-16) + 4 = 64 + 4 = 68  The third term matches the given sequence (68), so this formula seems to be correct. Let's continue to verify with the fourth term.  a_(4) = -4 * a_(3) + 4 = -4 * 68 + 4 = -272 + 4 = -268  The fourth term is -268, which continues the pattern of the sequence. Therefore, this formula is correct.Test Formula -3: We do not need to test the remaining options because we have already found the correct formula. However, for completeness, let's quickly check the third option.  Testing the recursive sequence formula a_(n)=-a_(n-1)-3 with a_(1)=5.  a_(2) = -a_(1) - 3 = -5 - 3 = -8  The second term does not match the given sequence (-16), so this formula is incorrect.Test Formula -4: Finally, we test the recursive sequence formula a_(n)=4a_(n-1)-4 with a_(1)=5.  a_(2) = 4 * a_(1) - 4 = 4 * 5 - 4 = 20 - 4 = 16  The second term does not match the given sequence (-16), so this formula is incorrect.

Which recursive sequence would produce the sequence $5,-16,68, \ldots$ ? $\newline$ $a_{1}=5$ and $a_{n}=-3 a_{n-1}-1$ $\newline$ $a_{1}=5$ and $a_{n}=-4 a_{n-1}+4$ $\newline$ $a_{1}=5$ and $a_{n}=-a_{n-1}-3$ $\newline$ $a_{1}=5$ and $a_{n}=4 a_{n-1}-4$

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