document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201437>Question 1201437</A>: <I><SPAN STYLE=\"word-break: break-all;\"> you and your parents are working on a scrapbook representing your family tree. He would like to include one page for the biography of each and sister, and your parents want you to include a page on yourself, classify the number of ancestors in each generation, as an arithmetic or geometric sequence.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835874 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=835874\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>Geometric sequence</font><BR>\n" );
document.write( "<font color=red>2^n</font> ancestors in generation n<BR>\n" );
document.write( "n = 1 is the previous generation, compared to your generation<BR>\n" );
document.write( "n = 2 is the previous two generations, etc\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Refer to this diagram to see a template of a simplified family tree<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=math_tutor2020&illustration_name=Screenshot_98.png\"><BR>\n" );
document.write( "The bottom of the family tree represents the present day and current generation.<BR>\n" );
document.write( "The youngest persons are always at the bottom, with the oldest at the top. <BR>\n" );
document.write( "As you move up the page, you move back in time.<BR>\n" );
document.write( "You can think of it like time flowing downhill, so going up the page reverses that flow.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As we move back through each generation, we double the number of people.<BR>\n" );
document.write( "This is simply from the obvious fact two parents are needed for any given individual.<BR>\n" );
document.write( "Therefore, we have a geometric sequence with common ratio r = 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We start with one person, then double to 2, that doubles to 4, and so on.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that's how we end up with 2^n, where n is the generation number mentioned earlier.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Interesting fact about powers of 2: <BR>\n" );
document.write( "Let's say we add you and your parents to get 1+2 = 3 people<BR>\n" );
document.write( "Now add on the four grand parents: 3+4 = 7<BR>\n" );
document.write( "Then the 8 great grand parents: 7+8 = 15<BR>\n" );
document.write( "Then the 16 great great grand parents: 15+16 = 31\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The subtotals were:<BR>\n" );
document.write( "3, 7, 15, 31\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add 1 to each of those<BR>\n" );
document.write( "3+1 = 4<BR>\n" );
document.write( "7+1 = 8<BR>\n" );
document.write( "15+1 = 16<BR>\n" );
document.write( "31+1 = 32<BR>\n" );
document.write( "We have powers of 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rule: the sum of the terms 1+2+4+8+...+2^n is 2^(n+1)-1<BR>\n" );
document.write( "I'll let you explore sums. Try out something like going back n = 6 generations to see how many total family members there are. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Keep in mind that this is a simplified family tree. By that I mean we make the assumption that each parenting couple has one child only. <BR>\n" );
document.write( "Realistically a family could have multiple children which would of course complicate things greatly. It would also break the geometric sequence property where the common ratio was r = 2. This is probably why your teacher will want the more simplified family version.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );