+586 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)?
+26367 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!
\(\mathbf{f(5) = -81}\)
