document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205379>Question 1205379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Cheyenne tells a joke to 5 people and those 5 people tell the joke to 5 more people, and so on. Which expression shows the number of people that will hear the joke after the sixth round?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842131 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=842131\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Open up your favorite spreadsheet program.<BR>\n" );
document.write( "I'm using LibreOffice Calc.\r<BR>\n" );
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document.write( "In cell A1 will go the label \"Round\" without quotes.<BR>\n" );
document.write( "B1 will have something like \"number who heard the joke\" without quotes.<BR>\n" );
document.write( "C1 will have \"Cumulative Sum\" without quotes.\r<BR>\n" );
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document.write( "Move to cell A2. <BR>\n" );
document.write( "Place the number 1 here. <BR>\n" );
document.write( "Then grab the bottom right corner of this cell to drag it down until the values 2,3,4,5,6 fill up cells A3 through A7.\r<BR>\n" );
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document.write( "Move to cell B2.<BR>\n" );
document.write( "Write the number 5 here.<BR>\n" );
document.write( "Move to cell B3. Type <font color=blue>=5*B2</font> into cell B3. <BR>\n" );
document.write( "The equal sign up front is important. This command marked in blue tells the spreadsheet \"multiply whatever is in B2 with the value 5\".<BR>\n" );
document.write( "The result of cell B3 should be 25.\r<BR>\n" );
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document.write( "Grab the bottom right corner of cell B3. Drag it down or doubleclick to populate the remaining values of column B. <BR>\n" );
document.write( "The values in column B, from top to bottom, should be: 5, 25, 125, 625, 3125, 15625\r<BR>\n" );
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document.write( "Now move to cell C2.<BR>\n" );
document.write( "Type <font color=blue>=B2</font> to copy the contents of B2 into C2.<BR>\n" );
document.write( "The equal sign is important.\r<BR>\n" );
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document.write( "In cell C3 type <font color=blue>=B3+C2</font> to add the contents of those cells. The result is 5+25 = 30<BR>\n" );
document.write( "Meaning there are 30 people total who heard the joke at the end of round 2 (the 5 from round 1, and 25 more from round 2, giving 5+25 = 30 total).<BR>\n" );
document.write( "Drag the bottom right corner of cell C3 down, or doubleclick to populate the rest of the values.<BR>\n" );
document.write( "The values in column C, from top to bottom, should be: 5, 30, 155, 780, 3905, <font color=red>19530</font>\r<BR>\n" );
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document.write( "This is what your spreadsheet should look like<BR>\n" );
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document.write( "<thead><tr><th>Round</th><th>number who heard joke</th><th>Cumulative Sum</th></tr></thead><tbody><tr><TD >1</TD><TD >5</TD><TD >5</TD></tr><tr><TD >2</TD><TD >25</TD><TD >30</TD></tr><tr><TD >3</TD><TD >125</TD><TD >155</TD></tr><tr><TD >4</TD><TD >625</TD><TD >780</TD></tr><tr><TD >5</TD><TD >3125</TD><TD >3905</TD></tr><tr><TD >6</TD><TD >15625</TD><TD ><font color=red>19530</font></TD></tr></tbody>\n" );
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document.write( "As a check, let's add up the values of everything in column B<BR>\n" );
document.write( "5 + 25 + 125 + 625 + 3125 + 15625 = <font color=red>19530</font>\r<BR>\n" );
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document.write( "There are a grand total of <font color=red>19530</font> people who have heard the joke after 6 rounds. \r<BR>\n" );
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document.write( "Note: This assumes that those who have heard it in any given round do NOT involve people from a previous round. <BR>\n" );
document.write( "Eg: if someone heard it in round 3, then they didn't hear it round 1 or round 2.\r<BR>\n" );
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document.write( "Alternative approach\r<BR>\n" );
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document.write( "The sequence 5, 25, 125, 625, 3125, 15625 is geometric.\r<BR>\n" );
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document.write( "a = 5 = first term<BR>\n" );
document.write( "r = 5 = common ratio<BR>\n" );
document.write( "a(n) = a*r^(n-1) = 5*5^(n-1) = number of people who heard the joke on round n<BR>\n" );
document.write( "where n is an integer and n = 1 is the starting index.\r<BR>\n" );
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document.write( "The expression 5*5^(n-1) is the number of people who heard the joke for that round only, and does not include previous rounds. <BR>\n" );
document.write( "To account for previous rounds, we'll need to use the next formula shown below.\r<BR>\n" );
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document.write( "Sn = sum of the first n terms of a geometric sequence<BR>\n" );
document.write( "Sn = a*(1 - r^n)/(1 - r)<BR>\n" );
document.write( "Sn = 5*(1 - 5^n)/(1 - 5)<BR>\n" );
document.write( "S6 = 5*(1 - 5^6)/(1 - 5)<BR>\n" );
document.write( "S6 = <font color=red>19530</font> is the sum of the first 6 terms of this geometric sequence.<BR>\n" );
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