Which recursive sequence would produce the sequence 3,-10,29,dots ? a_(1)=3 and a_(n)=-a_(n-1)-3 a_(1)=3 and a_(n)=-2a_(n-1)-4 a_(1)=3 and a_(n)=-4a_(n-1)-2 a_(1)=3 and a_(n)=-3a_(n-1)-1

Question

Which recursive sequence would produce the sequence  3,-10,29,dots ?  a_(1)=3 and  a_(n)=-a_(n-1)-3  a_(1)=3 and  a_(n)=-2a_(n-1)-4  a_(1)=3 and  a_(n)=-4a_(n-1)-2  a_(1)=3 and  a_(n)=-3a_(n-1)-1

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Test First Formula: Let's test each recursive formula given with the initial value a_(1)=3 to see which one produces the sequence 3, -10, 29, ... .Test Second Formula: First, we test the recursive formula a_(n)=-a_(n-1)-3 with a_(1)=3. Calculate a_(2) using the formula: a_(2)=-a_(1)-3 = -3-3 = -6. This does not match the second term of the sequence, which is -10.Test Third Formula: Since a_(2) does not match the second term of the sequence, we can eliminate the first recursive formula. Now, let's test the second recursive formula a_(n)=-2a_(n-1)-4 with a_(1)=3. Calculate a_(2) using the formula: a_(2)=-2a_(1)-4 = -2*3-4 = -6-4 = -10. This matches the second term of the sequence.Test Fourth Formula: Now, calculate a_(3) using the second recursive formula: a_(3)=-2a_(2)-4 = -2*(-10)-4 = 20-4 = 16. This does not match the third term of the sequence, which is 29.Conclusion: Since a_(3) does not match the third term of the sequence, we can eliminate the second recursive formula. Now, let's test the third recursive formula a_(n)=-4a_(n-1)-2 with a_(1)=3. Calculate a_(2) using the formula: a_(2)=-4a_(1)-2 = -4*3-2 = -12-2 = -14. This does not match the second term of the sequence, which is -10.Since a_(2) does not match the second term of the sequence, we can eliminate the third recursive formula. Now, let's test the fourth recursive formula a_(n)=-3a_(n-1)-1 with a_(1)=3. Calculate a_(2) using the formula: a_(2)=-3a_(1)-1 = -3*3-1 = -9-1 = -10. This matches the second term of the sequence.Now, calculate a_(3) using the fourth recursive formula: a_(3)=-3a_(2)-1 = -3*(-10)-1 = 30-1 = 29. This matches the third term of the sequence.Since the fourth recursive formula a_(n)=-3a_(n-1)-1 with a_(1)=3 produces the correct second and third terms of the sequence, we can conclude that this is the correct recursive formula for the given sequence.

Which recursive sequence would produce the sequence $3,-10,29, \ldots$ ? $\newline$ $a_{1}=3$ and $a_{n}=-a_{n-1}-3$ $\newline$ $a_{1}=3$ and $a_{n}=-2 a_{n-1}-4$ $\newline$ $a_{1}=3$ and $a_{n}=-4 a_{n-1}-2$ $\newline$ $a_{1}=3$ and $a_{n}=-3 a_{n-1}-1$

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