+619 The sequence $x_1$, $x_2$, $x_3$, . . ., has the property that $x_n = x_{n - 1} + x_{n - 2}$ for all $n \ge 3$. If $x_{11} - x_1 = 99$, then determine $x_6.$
+26393 The sequence x_1, x_2, x_3, . . ., has the property that x_n = x_{n - 1} + x_{n - 2}
for all n \ge 3.
If x_{11} - x_1 = 99, then determine x_6.
\(\begin{array}{|rcll|} \hline x_3 &=& x_2 + x_1 &=& 1\cdot x_2 + 1\cdot x_1 \\ x_4 &=& x_3 + x_2 &=& 2\cdot x_2 + 1\cdot x_1 \\ x_5 &=& x_4 + x_3 &=& 3\cdot x_2 + 2\cdot x_1 \\ \color{red}x_6 &\color{red}=& x_5 + x_4 &=& \color{red}5\cdot x_2 + 3\cdot x_1 \\ x_7 &=& x_6 + x_5 &=& 8\cdot x_2 + 5\cdot x_1 \\ x_8 &=& x_7 + x_6 &=& 13\cdot x_2 + 8\cdot x_1 \\ x_9 &=& x_8 + x_7 &=& 21\cdot x_2 + 13\cdot x_1 \\ x_{10} &=& x_9 + x_8 &=& 34\cdot x_2 + 21\cdot x_1 \\\\ x_{11} &=& x_{10} + x_9 &=& 55\cdot x_2 + 34\cdot x_1 \quad & | \quad x_{11} = 99 + x_1 \\ 99 + x_1 &=& 55\cdot x_2 + 34\cdot x_1 && \quad & | \quad - x_1 \\ 99 &=& 55\cdot x_2 + 33\cdot x_1 && \quad & | \quad : 11 \\ \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 \\\\ \hline \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 && \quad & | \quad \color{red}x_6 = 5\cdot x_2 + 3\cdot x_1\\ \mathbf{9} &\mathbf{=}& \mathbf{x_6} \\ \hline \end{array} \)
+129852 \($x_n = x_{n - 1} + x_{n - 2}$ for all $n \ge 3$\)
x11 = 99 + x1
x9 + x10 = 99 + (x3 - x2)
(x7 + x8) + (x8 + x9) = 99 + (x5 - x4) - (x3 - x1)
(x5 + x6) + (x6 + x7) + (x6 + x7) + (x7 + x8) = 99 + x5 - x4 - x3 + x1
3x6 + 3x7 + x8 = 99 - x4 - x3 + x1
3x6 + 3(x5 + x6) + (x6 + x7) = 99 - (x6 - x5) - ( x5 - x4) + (x3 - x2)
7x6 + 3x5 + x7 = 99 -x6 +x4 + x3 - x2
8x6 + 3(x7 - x6) + x7 = 99 + x4 +( x5 - x4) - (x4 - x3)
5x6 + 4x7 = 99 + x5 - x4 + x3
5x6 + 4(x5 + x6) = 99 + x5 - (x6 - x5) + (x5 - x4)
10x6 + 4x5 = 99 + 3x5 -x4
10x6 + x5 = 99 - x4
10x6 + x5 + x4 = 99
10x6 + x6 = 99
11x6 = 99 divide both sides by 11
x6 = 9
+26393 The sequence x_1, x_2, x_3, . . ., has the property that x_n = x_{n - 1} + x_{n - 2}
for all n \ge 3.
If x_{11} - x_1 = 99, then determine x_6.
\(\begin{array}{|rcll|} \hline x_3 &=& x_2 + x_1 &=& 1\cdot x_2 + 1\cdot x_1 \\ x_4 &=& x_3 + x_2 &=& 2\cdot x_2 + 1\cdot x_1 \\ x_5 &=& x_4 + x_3 &=& 3\cdot x_2 + 2\cdot x_1 \\ \color{red}x_6 &\color{red}=& x_5 + x_4 &=& \color{red}5\cdot x_2 + 3\cdot x_1 \\ x_7 &=& x_6 + x_5 &=& 8\cdot x_2 + 5\cdot x_1 \\ x_8 &=& x_7 + x_6 &=& 13\cdot x_2 + 8\cdot x_1 \\ x_9 &=& x_8 + x_7 &=& 21\cdot x_2 + 13\cdot x_1 \\ x_{10} &=& x_9 + x_8 &=& 34\cdot x_2 + 21\cdot x_1 \\\\ x_{11} &=& x_{10} + x_9 &=& 55\cdot x_2 + 34\cdot x_1 \quad & | \quad x_{11} = 99 + x_1 \\ 99 + x_1 &=& 55\cdot x_2 + 34\cdot x_1 && \quad & | \quad - x_1 \\ 99 &=& 55\cdot x_2 + 33\cdot x_1 && \quad & | \quad : 11 \\ \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 \\\\ \hline \color{red}9 &\color{red}=& \color{red}5\cdot x_2 + 3\cdot x_1 && \quad & | \quad \color{red}x_6 = 5\cdot x_2 + 3\cdot x_1\\ \mathbf{9} &\mathbf{=}& \mathbf{x_6} \\ \hline \end{array} \)
