document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209400>Question 1209400</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a_1, a_2, a_3, \dots be a sequence. If<BR>\n" );
document.write( "a_n = a_{n - 1} + a_{n - 2}<BR>\n" );
document.write( "for all n \ge 3, and a_{11} = 4 and a_{10} = 1, then find a_6. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848753 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848753\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a<sub>3</sub> = a<sub>2</sub>+a<sub>1</sub><BR>\n" );
document.write( "a<sub>4</sub> = a<sub>3</sub>+a<sub>2</sub><BR>\n" );
document.write( "a<sub>5</sub> = a<sub>4</sub>+a<sub>3</sub><BR>\n" );
document.write( "a<sub>6</sub> = a<sub>5</sub>+a<sub>4</sub><BR>\n" );
document.write( "... etc ... <BR>\n" );
document.write( "a<sub>11</sub> = a<sub>10</sub>+a<sub>9</sub><BR>\n" );
document.write( "The indices decrease when reading left to right.\r<BR>\n" );
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document.write( "Let's rearrange each equation so that the last term on the right hand side is by itself.<BR>\n" );
document.write( "a<sub>1</sub> = a<sub>3</sub>-a<sub>2</sub><BR>\n" );
document.write( "a<sub>2</sub> = a<sub>4</sub>-a<sub>3</sub><BR>\n" );
document.write( "a<sub>3</sub> = a<sub>5</sub>-a<sub>4</sub><BR>\n" );
document.write( "a<sub>4</sub> = a<sub>6</sub>-a<sub>5</sub><BR>\n" );
document.write( "... etc ...<BR>\n" );
document.write( "a<sub>9</sub> = a<sub>11</sub>-a<sub>10</sub><BR>\n" );
document.write( "The template is a_n = a_{n+2} - a_{n+1}\r<BR>\n" );
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document.write( "Since we are given a<sub>11</sub> = 4 and a<sub>10</sub> = 1, we can then say,<BR>\n" );
document.write( "a<sub>9</sub> = a<sub>11</sub>-a<sub>10</sub><BR>\n" );
document.write( "a<sub>9</sub> = 4-1<BR>\n" );
document.write( "a<sub>9</sub> = 3<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "a<sub>8</sub> = a<sub>10</sub>-a<sub>9</sub><BR>\n" );
document.write( "a<sub>8</sub> = 1-3<BR>\n" );
document.write( "a<sub>8</sub> = -2<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "a<sub>7</sub> = a<sub>9</sub>-a<sub>8</sub><BR>\n" );
document.write( "a<sub>7</sub> = 3-(-2)<BR>\n" );
document.write( "a<sub>7</sub> = 3+2<BR>\n" );
document.write( "a<sub>7</sub> = 5<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "a<sub>6</sub> = a<sub>8</sub>-a<sub>7</sub><BR>\n" );
document.write( "a<sub>6</sub> = -2-5<BR>\n" );
document.write( "a<sub>6</sub> = <font color=red>-7</font>\r<BR>\n" );
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document.write( "There might be a faster more efficient approach.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );