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The sequence x_1x_2x_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.

Guest Jan 31, 2015

Best Answer 

 #1
avatar+18827 
+10

The sequence x_1x_2x_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.

$$\small{\text{
$
\begin{array}{rcrcrcr}
x_{11}&=& x_{10}+x_{9} &=& (x_9+x_8)+x_9 &=& 2x_9+x_8\\
&=& 2x_9+x_8 &=& 2(x_8+x_7)+x_8 &=& 3x_8+2x_7\\
&=& 3x_8+2x_7 &=& 3(x_7+x_6)+2x_7 &=& 5x_7+3x_6\\
&=& 5x_7+3x_6 &=& 5(x_6+x_5)+3x_6 &=& 8x_6+5x_5\\
&=& 8x_6+5x_5 &=& 8(x_5+x_4)+5x_5 &=& 13x_5+8x_4\\
&=& 13x_5+8x_4 &=& 13(x_4+x_3)+8x_4 &=& 21x_4+13x_3\\
&=& 21x_4+13x_3 &=& 21(x_3+x_2)+13x_3 &=& 34x_3+21x_2\\
&=& 34x_3+21x_2 &=& 34(x_2+x_1)+21x_2 &=& 55x_2+34x_1\\
\end{array}
$
}}\\
\small{\text{$x_{11} = 55x_2+34x_1$}}$$

x_{11} - x_1 = 99 

$$\small{\text{$
\begin{array}{rcl}
55x_2+34x_1 -x_1 &=& 99 \\
55x_2 +33x_1&=&99 \quad | \quad :11 \\
\boxed{5x_2+3x_1 = 9}
\end{array}
$
}}$$

$$\small{\text{
$
\begin{array}{rcrcr}
x_3 &=& &=& x_2+x_1 \\
x_4 &=& x_3+x_2 &=& 2x_2+x_1 \\
x_5 &=& x_4+x_3 &=& 3x_2+2x_1 \\
x_6 &=& x_5+x_4 &=& 5x_2+3x_1 \\
\end{array}
$
}}\\
\small{\text{$x_6= 5x_2+3x_1 $}}=9$$

heureka  Feb 1, 2015
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2+0 Answers

 #1
avatar+18827 
+10
Best Answer

The sequence x_1x_2x_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.

$$\small{\text{
$
\begin{array}{rcrcrcr}
x_{11}&=& x_{10}+x_{9} &=& (x_9+x_8)+x_9 &=& 2x_9+x_8\\
&=& 2x_9+x_8 &=& 2(x_8+x_7)+x_8 &=& 3x_8+2x_7\\
&=& 3x_8+2x_7 &=& 3(x_7+x_6)+2x_7 &=& 5x_7+3x_6\\
&=& 5x_7+3x_6 &=& 5(x_6+x_5)+3x_6 &=& 8x_6+5x_5\\
&=& 8x_6+5x_5 &=& 8(x_5+x_4)+5x_5 &=& 13x_5+8x_4\\
&=& 13x_5+8x_4 &=& 13(x_4+x_3)+8x_4 &=& 21x_4+13x_3\\
&=& 21x_4+13x_3 &=& 21(x_3+x_2)+13x_3 &=& 34x_3+21x_2\\
&=& 34x_3+21x_2 &=& 34(x_2+x_1)+21x_2 &=& 55x_2+34x_1\\
\end{array}
$
}}\\
\small{\text{$x_{11} = 55x_2+34x_1$}}$$

x_{11} - x_1 = 99 

$$\small{\text{$
\begin{array}{rcl}
55x_2+34x_1 -x_1 &=& 99 \\
55x_2 +33x_1&=&99 \quad | \quad :11 \\
\boxed{5x_2+3x_1 = 9}
\end{array}
$
}}$$

$$\small{\text{
$
\begin{array}{rcrcr}
x_3 &=& &=& x_2+x_1 \\
x_4 &=& x_3+x_2 &=& 2x_2+x_1 \\
x_5 &=& x_4+x_3 &=& 3x_2+2x_1 \\
x_6 &=& x_5+x_4 &=& 5x_2+3x_1 \\
\end{array}
$
}}\\
\small{\text{$x_6= 5x_2+3x_1 $}}=9$$

heureka  Feb 1, 2015
 #2
avatar+80913 
+3

Very nice, heureka......the additions on the right hand side are numbers in the Fibonacci series..!!!

 

CPhill  Feb 1, 2015

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