Which recursive sequence would produce the sequence 5,14,32,dots ? a_(1)=5 and a_(n)=4a_(n-1)+2 a_(1)=5 and a_(n)=-a_(n-1)+3 a_(1)=5 and a_(n)=2a_(n-1)+4 a_(1)=5 and a_(n)=3a_(n-1)-1

Question

Which recursive sequence would produce the sequence  5,14,32,dots ?  a_(1)=5 and  a_(n)=4a_(n-1)+2  a_(1)=5 and  a_(n)=-a_(n-1)+3  a_(1)=5 and  a_(n)=2a_(n-1)+4  a_(1)=5 and  a_(n)=3a_(n-1)-1

Bytelearn · Accepted Answer

Calculate Second Term: To determine which recursive sequence produces the given sequence, we will calculate the second term using each of the provided recursive formulas and compare it to the second term of the given sequence, which is 14.Try Formula 1: First, let's try the recursive formula a_(1)=5 and a_(n)=4a_(n-1)+2. We start with a_(1)=5 and calculate a_(2). a_(2) = 4a_(1) + 2 = 4*5 + 2 = 20 + 2 = 22. This does not match the second term of the given sequence, which is 14.Try Formula 2: Next, let's try the recursive formula a_(1)=5 and a_(n)=-a_(n-1)+3. We start with a_(1)=5 and calculate a_(2). a_(2) = -a_(1) + 3 = -5 + 3 = -2. This does not match the second term of the given sequence, which is 14.Try Formula 3: Now, let's try the recursive formula a_(1)=5 and a_(n)=2a_(n-1)+4. We start with a_(1)=5 and calculate a_(2). a_(2) = 2a_(1) + 4 = 2*5 + 4 = 10 + 4 = 14. This matches the second term of the given sequence, which is 14. However, we need to verify that the third term also matches before we can conclude this is the correct formula.Verify Formula 3: Using the same formula a_(1)=5 and a_(n)=2a_(n-1)+4, let's calculate a_(3). a_(3) = 2a_(2) + 4 = 2*14 + 4 = 28 + 4 = 32. This matches the third term of the given sequence, which is 32. Therefore, this recursive formula produces the given sequence.Check Formula 4: Finally, for completeness, let's check the last recursive formula a_(1)=5 and a_(n)=3a_(n-1)-1. We start with a_(1)=5 and calculate a_(2). a_(2) = 3a_(1) - 1 = 3*5 - 1 = 15 - 1 = 14. This matches the second term of the given sequence, but we need to check the third term as well.Check Formula 4: Using the formula a_(1)=5 and a_(n)=3a_(n-1)-1, let's calculate a_(3). a_(3) = 3a_(2) - 1 = 3*14 - 1 = 42 - 1 = 41. This does not match the third term of the given sequence, which is 32. Therefore, this recursive formula does not produce the given sequence.

Which recursive sequence would produce the sequence $5,14,32, \ldots$ ? $\newline$ $a_{1}=5$ and $a_{n}=4 a_{n-1}+2$ $\newline$ $a_{1}=5$ and $a_{n}=-a_{n-1}+3$ $\newline$ $a_{1}=5$ and $a_{n}=2 a_{n-1}+4$ $\newline$ $a_{1}=5$ and $a_{n}=3 a_{n-1}-1$

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