Which recursive sequence would produce the sequence 2,-7,38,dots? a_(1)=2 and a_(n)=5a_(n-1)-6 a_(1)=2 and a_(n)=3a_(n-1)-5 a_(1)=2 and a_(n)=-5a_(n-1)+3 a_(1)=2 and a_(n)=-6a_(n-1)+5

Question

Which recursive sequence would produce the sequence  2,-7,38,dots?  a_(1)=2 and  a_(n)=5a_(n-1)-6  a_(1)=2 and  a_(n)=3a_(n-1)-5  a_(1)=2 and  a_(n)=-5a_(n-1)+3  a_(1)=2 and  a_(n)=-6a_(n-1)+5

Bytelearn · Accepted Answer

Initialize and Test First Option: To determine which recursive sequence produces the given sequence, we will apply each given recursive formula to the initial value a_(1)=2 and see which one generates the sequence 2, -7, 38, ...Test Second Option: Let's start with the first option: a_(1)=2 and a_(n)=5a_(n-1)-6. We will calculate a_(2) using this formula. a_(2) = 5a_(1) - 6 = 5*2 - 6 = 10 - 6 = 4 This does not match the second term of the sequence, which is -7, so this option is incorrect.Test Third Option: Now let's try the second option: a_(1)=2 and a_(n)=3a_(n-1)-5. We will calculate a_(2) using this formula. a_(2) = 3a_(1) - 5 = 3*2 - 5 = 6 - 5 = 1 This does not match the second term of the sequence, which is -7, so this option is also incorrect.Confirm Third Option: Next, we will test the third option: a_(1)=2 and a_(n)=-5a_(n-1)+3. We will calculate a_(2) using this formula. a_(2) = -5a_(1) + 3 = -5*2 + 3 = -10 + 3 = -7 This matches the second term of the sequence, so this option could be correct. We need to calculate a_(3) to confirm.Test Fourth Option: Using the third option's formula, we calculate a_(3). a_(3) = -5a_(2) + 3 = -5*(-7) + 3 = 35 + 3 = 38 This matches the third term of the sequence, so this option is indeed correct.Finalize: Finally, for completeness, let's check the fourth option: a_(1)=2 and a_(n)=-6a_(n-1)+5. We will calculate a_(2) using this formula. a_(2) = -6a_(1) + 5 = -6*2 + 5 = -12 + 5 = -7 This matches the second term of the sequence, so we need to calculate a_(3) to see if this option could also be correct.Using the fourth option's formula, we calculate a_(3). a_(3) = -6a_(2) + 5 = -6*(-7) + 5 = 42 + 5 = 47 This does not match the third term of the sequence, which is 38, so this option is incorrect.

Which recursive sequence would produce the sequence $2,-7,38, \ldots ?$ $\newline$ $a_{1}=2$ and $a_{n}=5 a_{n-1}-6$ $\newline$ $a_{1}=2$ and $a_{n}=3 a_{n-1}-5$ $\newline$ $a_{1}=2$ and $a_{n}=-5 a_{n-1}+3$ $\newline$ $a_{1}=2$ and $a_{n}=-6 a_{n-1}+5$

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