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The function f(n) = 3f(n-2) - 2f(n-1), where f(2) = 3 and f(1) = -1. What is the value of f(5)?

 Aug 16, 2018
 #1
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The function f(n) = 3f(n-2) - 2f(n-1), where f(2) = 3 and f(1) = -1. What is the value of f(5)?

 

Formula:

\(\begin{array}{|rcll|} \hline f_n &=& -2f_{n-1}+3f_{n-2} \quad & | \quad f_2 = 3 \qquad f_1 = -1 \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline f_3 &=& -2f_{2}+3f_{1} \quad & | \quad f_2 = 3 \qquad f_1 = -1 \\ &=& -2(3)+3(-1) \\ &=& -6-3 \\ \mathbf{f_3} & \mathbf{=} & \mathbf{-9} \\\\ \hline f_4 &=& -2f_{3}+3f_{2} \quad & | \quad f_3 = -9 \qquad f_2 = 3 \\ &=& -2(-9)+3(3) \\ &=& 18+9 \\ \mathbf{f_4} & \mathbf{=} & \mathbf{27} \\\\ \hline f_5 &=& -2f_{4}+3f_{3} \quad & | \quad f_4 = 27 \qquad f_3 = -9 \\ &=& -2(27)+3(-9) \\ &=& -54 -27 \\ \mathbf{f_5} & \mathbf{=} & \mathbf{-81} \\ \hline \end{array}\)

 

\(\mathbf{f(5) = -81}\)

 

Formula:

\(\begin{array}{|lrcll|} \hline & f_n &=& -2f_{n-1}+3f_{n-2} \\ \text{ or } \\ & f_n &=& -(-3)^{n-1} \\ & f_5 &=& -(-3)^{5-1} \\ & f_5 &=& -(-3)^{4} \\ & \mathbf{f_5} & \mathbf{=} & \mathbf{-81} \\ \hline \end{array}\)

Formula edited, thank you Alan!

 

\(\mathbf{f(5) = -81}\)

 

laugh

 Aug 16, 2018
edited by heureka  Aug 17, 2018
edited by heureka  Aug 17, 2018
edited by heureka  Aug 17, 2018
edited by heureka  Aug 17, 2018
edited by heureka  Aug 17, 2018
edited by heureka  Aug 17, 2018

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