document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199189>Question 1199189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bus driver who starts with a large empty bus picks us 11 riders at every odd- numbered stop and drops off 5 riders at every even-numbered stop. The stop after which the bus will first contain more than 165 passengers is stop number<BR>\n" );
document.write( " a) 27 b) 28 c) 52 d) 53 e) 55 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832920 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=832920\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The sequence<BR>\n" );
document.write( "11,6,17,12,23,18,...<BR>\n" );
document.write( "represents the number of riders on the bus for any particular stop.\r<BR>\n" );
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document.write( "We start with 11 riders on stop 1 (odd number).<BR>\n" );
document.write( "Then the bus drops off 5 riders on stop 2 (even number) to have 11-5 = 6 people remaining.\r<BR>\n" );
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document.write( "For the next odd numbered stop, stop 3, there are 11 more riders to have 6+11 = 17 people.<BR>\n" );
document.write( "For the next even numbered stop, stop 4, exactly 5 people leave to have 17-5 = 12 people remaining.\r<BR>\n" );
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document.write( "This process continues until the bus is full of course, or when there aren't any other stops to make. <BR>\n" );
document.write( "We'll assume that the bus can hold 165 people or a bit more.\r<BR>\n" );
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document.write( "Going back to this sequence:<BR>\n" );
document.write( "11,6,17,12,23,18,...\r<BR>\n" );
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document.write( "Let's highlight the even numbered terms.<BR>\n" );
document.write( "11,<font color=blue>6</font>,17,<font color=blue>12</font>,23,<font color=blue>18</font>,...<BR>\n" );
document.write( "This is the set of multiples of 6.<BR>\n" );
document.write( "Since 2 is a factor of 6, each item is even.<BR>\n" );
document.write( "Therefore, it is impossible to have 165 people on the bus for any even numbered stop since 165 is odd. \r<BR>\n" );
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document.write( "Let's highlight the odd numbered terms.<BR>\n" );
document.write( "<font color=red>11</font>,6,<font color=red>17</font>,12,<font color=red>23</font>,18,...<BR>\n" );
document.write( "Notice that we're increasing by 6 each time. <BR>\n" );
document.write( "This is exactly because 11-5 = 6 is the net change in the number of people on the bus. \r<BR>\n" );
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document.write( "Therefore, the odd numbered terms of this sequence are arithmetic<BR>\n" );
document.write( "a = 11 = first term<BR>\n" );
document.write( "d = 6 = common difference<BR>\n" );
document.write( "a(n) = a+d*(n-1) = nth term<BR>\n" );
document.write( "a(n) = 11+6*(n-1)\r<BR>\n" );
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document.write( "Let's see if we get 165 in this sequence<BR>\n" );
document.write( "a(n) = 11+6*(n-1)<BR>\n" );
document.write( "165 = 11+6*(n-1)<BR>\n" );
document.write( "165 = 11+6n-6<BR>\n" );
document.write( "165 = 6n+5<BR>\n" );
document.write( "6n+5 = 165<BR>\n" );
document.write( "6n = 165-5<BR>\n" );
document.write( "6n = 160<BR>\n" );
document.write( "n = 160/6<BR>\n" );
document.write( "n = 26.667<BR>\n" );
document.write( "We don't have 165 in the odd numbered terms since n must be a positive integer.<BR>\n" );
document.write( "But it gives us an idea where the min number over 165 could be located.\r<BR>\n" );
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document.write( "Let's revisit these highlighted terms:<BR>\n" );
document.write( "<font color=red>11</font>,6,<font color=red>17</font>,12,<font color=red>23</font>,18,...<BR>\n" );
document.write( "The \"11\" is the 1st odd number and 1st term overall.<BR>\n" );
document.write( "The \"17\" is the 2nd odd number and 3rd term overall.<BR>\n" );
document.write( "The \"23\" is the 3rd odd number and 5th term overall.<BR>\n" );
document.write( "And so on.<BR>\n" );
document.write( "For the k<sup>th</sup> odd numbered term, it refers to the (2k-1)<sup>th</sup> term overall, where k is a positive integer.\r<BR>\n" );
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document.write( "If n = 26, then,<BR>\n" );
document.write( "a(n) = 11+6*(n-1)<BR>\n" );
document.write( "a(26) = 11+6*(26-1)<BR>\n" );
document.write( "a(26) = 161<BR>\n" );
document.write( "The 26th odd numbered bus stop will have 161 people on the bus. <BR>\n" );
document.write( "We haven't gone over 165, but we're really close.\r<BR>\n" );
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document.write( "If n = 27, then,<BR>\n" );
document.write( "a(n) = 11+6*(n-1)<BR>\n" );
document.write( "a(27) = 11+6*(27-1)<BR>\n" );
document.write( "a(27) = 167<BR>\n" );
document.write( "We now have gone over 165.<BR>\n" );
document.write( "This is the smallest such n to achieve the goal.\r<BR>\n" );
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document.write( "The k = 27 odd numbered term refers to the 2k-1 = 2*27-1 = 54-1 = 53rd term overall of the sequence 11,6,17,12,23,18,...\r<BR>\n" );
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document.write( "Answer: <font color=red>53 (choice D)</font><BR>\n" );
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