document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972444>Question 972444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If none of your ancestors appears in more than one line of descent, how many ancestors have you had in the generations since the discovery of America in 1942 by Columbus? Assume that each generation covers approximately 33 years.<BR>\n" );
document.write( "(We are currently studying Mathematical Induction & Progressions) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #594867 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=594867\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It will depend en when you were born, but I assume you are between the ages of 10 and 25.<BR>\n" );
document.write( "I also understand the question as asking how many of your ancestors were born since the discovery of America in 1942. At that time, some ancestor that had been born before could still be alive, but they are not counted.<BR>\n" );
document.write( "Between October 1492 and today (May 2015), the number of years passed is almost<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2015-1492=523 ALIGN=MIDDLE  ALT=\"2015-1492=523\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If we assume each generation takes 33 years (meaning each person is born to parents that are 33 years old),<BR>\n" );
document.write( "there have been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=523%2F33=15.85 ALIGN=MIDDLE  ALT=\"523%2F33=15.85\"   DISABLED_style= \"max-width:90%; height:auto;\" > generations since the discovery of America.<BR>\n" );
document.write( "So if you were born in the year 2000,<BR>\n" );
document.write( "your parents (the first generation of your ancestors, counting away from you) were born in the year<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=2000-1%2A33 ALIGN=MIDDLE  ALT=\"a%5B1%5D=2000-1%2A33\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Your grandparents were born in the year<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=2000-2%2A33 ALIGN=MIDDLE  ALT=\"a%5B2%5D=2000-2%2A33\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you think that looks like an arithmetic progression, you are right.<BR>\n" );
document.write( "(If you live in the USA, and think that is an arithmetic sequence, you are right too.<BR>\n" );
document.write( "Whatever you call it, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values keep happening with a common difference of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-33 ALIGN=MIDDLE  ALT=\"-33\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "all the way up (or down) to the generation born in the year<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B15%5D=2000-15%2A33=2000-495=1505 ALIGN=MIDDLE  ALT=\"a%5B15%5D=2000-15%2A33=2000-495=1505\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So if you were born around the year 2000 (give or take about a decade),<BR>\n" );
document.write( "you have 15 generations of ancestors that were born after the discovery of America in 1942.<BR>\n" );
document.write( "Now comes the geometric progression (or sequence).<BR>\n" );
document.write( "In the generation of your parents, you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D=2 ALIGN=MIDDLE  ALT=\"b%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ancestors: your mother and father.<BR>\n" );
document.write( "In the generation before that, there were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ancestors for each of your parents,<BR>\n" );
document.write( "for a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D=2%2A2=2%5E2=4 ALIGN=MIDDLE  ALT=\"b%5B2%5D=2%2A2=2%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ancestors: your <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > grandparents.<BR>\n" );
document.write( "In the generation before that, you had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B3%5D=2%5E3=8 ALIGN=MIDDLE  ALT=\"b%5B3%5D=2%5E3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > great grandparents,<BR>\n" );
document.write( "and so on.<BR>\n" );
document.write( "In the first generation of your ancestors born after the discovery of America in 1942,<BR>\n" );
document.write( "there were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B15%5D=2%5E15 ALIGN=MIDDLE  ALT=\"b%5B15%5D=2%5E15\"   DISABLED_style= \"max-width:90%; height:auto;\" > people,<BR>\n" );
document.write( "unless one (or more) of your ancestors appears in more that one line of descent,<BR>\n" );
document.write( "as in the case of one of my great-great-great-grandmothers,<BR>\n" );
document.write( "who had two of her grandchildren marry each other.<BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D ALIGN=MIDDLE  ALT=\"b%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values form a geometric progression (or sequence)<BR>\n" );
document.write( "whose first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D=2 ALIGN=MIDDLE  ALT=\"b%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and each successive term is the previous one times the common ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2 ALIGN=MIDDLE  ALT=\"r=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29=2%2A2%5E%28n-1%29=2%5En ALIGN=MIDDLE  ALT=\"b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29=2%2A2%5E%28n-1%29=2%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total number of your ancestors born after the discovery of America in 1942 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B15%5D=b%5B1%5D%2Bb%5B2%5D%2Bb%5B3%5D%2Bellipsis%2Bb%5B15%5D ALIGN=MIDDLE  ALT=\"S%5B15%5D=b%5B1%5D%2Bb%5B2%5D%2Bb%5B3%5D%2Bellipsis%2Bb%5B15%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you either memorized (or can look up) the formula for a geometric series,<BR>\n" );
document.write( "you would know that in this case<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B15%5D=2%5E16-2=65534 ALIGN=MIDDLE  ALT=\"S%5B15%5D=2%5E16-2=65534\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The pesky formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=b%5B1%5D%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=b%5B1%5D%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but you do not need to memorize that if you know how to derive it,<BR>\n" );
document.write( "and deriving it for this particular case makes mistakes less likely:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B15%5D=2%2B2%5E2%2Bellipsis%2B2%5E14%2B2%5E15 ALIGN=MIDDLE  ALT=\"S%5B15%5D=2%2B2%5E2%2Bellipsis%2B2%5E14%2B2%5E15\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2AS%5B15%5D=2%5E2%2B2%5E3%2Bellipsis%2B2%5E15%2B2%5E16 ALIGN=MIDDLE  ALT=\"2%2AS%5B15%5D=2%5E2%2B2%5E3%2Bellipsis%2B2%5E15%2B2%5E16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B15%5D=%282-1%29%2AS%5B15%5D=2%2AS%5B15%5D-S%5B15%5D=2%5E16-2 ALIGN=MIDDLE  ALT=\"S%5B15%5D=%282-1%29%2AS%5B15%5D=2%2AS%5B15%5D-S%5B15%5D=2%5E16-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );