document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150903>Question 1150903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Vanessa, the student council president, needed to get a message to the whole school, so she set up an email tree. She sent the message to the two vice presidents and made them forward it to two other students. Suppose the pattern is repeated and no student receives it more than once!\r<BR>\n" );
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document.write( "a) How many people receive it during the seventh mailing?<BR>\n" );
document.write( "b) Which mailing will be sent to 500 people?<BR>\n" );
document.write( "c) There are 500 students in the school, so how many does it take to send the mail to all of them? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772468 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=772468\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Part a)<BR>\n" );
document.write( "During the nth mailing, 2^n students will be messaged. So for the first mailing (n=1) we have 2^n = 2^1 = 2 students messaged (the two vice presidents). Then for round two (n=2), we have 2^n = 2^2 = 4 students messaged in this round.\r<BR>\n" );
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document.write( "For the 7th mailing, plug n = 7 into 2^n to get 2^n = 2^7 = 128. \r<BR>\n" );
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document.write( "Answer: <font color=red size=4>128</font>\r<BR>\n" );
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document.write( "Part b)\r<BR>\n" );
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document.write( "Think backwards from part a). We want to aim for 500 students in a single round of messaging. The question is \"what value of n will make 2^n equal 500?\". \r<BR>\n" );
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document.write( "We want to solve<BR>\n" );
document.write( "2^n = 500\r<BR>\n" );
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document.write( "Use logarithms to answer that question. You will need your calculator.<BR>\n" );
document.write( "2^n = 500<BR>\n" );
document.write( "Log(2^n) = Log(500)<BR>\n" );
document.write( "n*Log(2) = Log(500)<BR>\n" );
document.write( "n = Log(500)/Log(2)<BR>\n" );
document.write( "n = 8.96578428466209\r<BR>\n" );
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document.write( "Note how if n = 8, then<BR>\n" );
document.write( "2^n = 2^8 = 256<BR>\n" );
document.write( "and when n = 9, we have<BR>\n" );
document.write( "2^n = 2^9 = 512<BR>\n" );
document.write( "so as expected, the solution is somewhere in between n = 8 and n = 9\r<BR>\n" );
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document.write( "Using n = 8 is too small, so we have to go with n = 9\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>9th round</font>\r<BR>\n" );
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document.write( "Part c)\r<BR>\n" );
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document.write( "Here is a list of powers of 2<BR>\n" );
document.write( "1,2,4,8,16,32,64,128,256,512,...\r<BR>\n" );
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document.write( "Let's compute partial sums<BR>\n" );
document.write( "1+2 = 3<BR>\n" );
document.write( "1+2+4 = 7<BR>\n" );
document.write( "1+2+4+8 = 15<BR>\n" );
document.write( "1+2+4+8+16 = 31<BR>\n" );
document.write( "1+2+4+8+16+32 = 63<BR>\n" );
document.write( "1+2+4+8+16+32+64 = 127<BR>\n" );
document.write( "1+2+4+8+16+32+64+128 = 255<BR>\n" );
document.write( "1+2+4+8+16+32+64+128+256 = 511<BR>\n" );
document.write( "Each sum is 1 less than the next power of 2. The general formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \sum_{k=0}^{n}2^k = 2^{n+1}-1\">\r<BR>\n" );
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document.write( "What this means is that we can quickly compute how many students have been messaged in total (rather than just on an individual round).\r<BR>\n" );
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document.write( "Set 2^(n+1) - 1 equal to 500 and solve for n.\r<BR>\n" );
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document.write( "2^(n+1) - 1 = 500<BR>\n" );
document.write( "2^(n+1) = 500+1<BR>\n" );
document.write( "2^(n+1) = 501<BR>\n" );
document.write( "Log(2^(n+1)) = Log(501)<BR>\n" );
document.write( "(n+1)*Log(2) = Log(501)<BR>\n" );
document.write( "n+1 = Log(501)/Log(2)<BR>\n" );
document.write( "n+1 = 8.9686667931952<BR>\n" );
document.write( "n = 8.9686667931952-1<BR>\n" );
document.write( "n = 7.9686667931952<BR>\n" );
document.write( "That rounds to n = 8\r<BR>\n" );
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document.write( "By round 8, a total of <BR>\n" );
document.write( "2^(n+1)-1 = 2^(8+1)-1 = 511<BR>\n" );
document.write( "people have been messaged<BR>\n" );
document.write( "If we ignore Vanessa, then 511-1 = 510 people have been messaged\r<BR>\n" );
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document.write( "Of course this school has 500 students and not 510 or 511. The same value of n is the answer however because in the 8th round 2^8 = 256 people are messaged. Subtracting 11 from this count is still a positive value.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>8th round</font>\r<BR>\n" );
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document.write( "As you can see, there is no need to have a 9th round in which 512 people are messaged because the entire school will be notified by the 8th round<BR>\n" );
document.write( "1+2+4+8+16+32+64+128+256 = 511<BR>\n" );
document.write( "note that 2^n = 2^8 = 256\r<BR>\n" );
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document.write( "</font> </SPAN>\n" );
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