Question

which recursive sequence would produce the sequence 5, -8, 18, ...?

answer attempt 1 out of 2

○ $a_1 = 5$ and $a_n = -a_{n - 1} - 3$
○ $a_1 = 5$ and $a_n = -3a_{n - 1} - 1$
○ $a_1 = 5$ and $a_n = -2a_{n - 1} + 2$
○ $a_1 = 5$ and $a_n = 2a_{n - 1} - 2$

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Explanation:

Step1: Test Option 1

Given \(a_1 = 5\), for \(n = 2\), \(a_2=-a_{1}-3=-5 - 3=-8\). For \(n = 3\), \(a_3=-a_{2}-3=-(-8)-3 = 8 - 3 = 5
eq18\). So Option 1 is wrong.

Step2: Test Option 2

Given \(a_1 = 5\), for \(n = 2\), \(a_2=-3a_{1}-1=-15 - 1=-16
eq - 8\). So Option 2 is wrong.

Step3: Test Option 3

Given \(a_1 = 5\), for \(n = 2\), \(a_2=-2a_{1}+2=-10 + 2=-8\). For \(n = 3\), \(a_3=-2a_{2}+2=-2\times(-8)+2 = 16 + 2 = 18\). This matches the sequence.

Step4: Test Option 4 (Optional)

Given \(a_1 = 5\), for \(n = 2\), \(a_2=2a_{1}-2 = 10 - 2 = 8
eq - 8\). So Option 4 is wrong.

Answer:

\(a_1 = 5\) and \(a_n=-2a_{n - 1}+2\) (the third option among the given choices)