document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099172>Question 1099172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following problem. Mike and Tim are looking to earn a little extra money. The beach committee offers them the opportunity off picking up plastic bottles, paying them $0.20 per bottle. Mike realizes as they pick up it will get harder and harder to find more bottles. So as an incentive to keep looking he suggest a different form of payment. He suggests $0.10 for the first bottle, and increase the pay by 2% for each bottle after that. Tim thinks Mike is crazy to propose an increase of just 2% per piece. He plans to ask for a one-cent increase for every piece, starting at 15 cents for the first bottle. Suppose the committee accepts each offer. <BR>\n" );
document.write( "x<BR>\n" );
document.write( "Answer the following in detail: <BR>\n" );
document.write( "For Mike and Tim, write the first 5 terms of a sequence for payment scheme for each piece of paper. <BR>\n" );
document.write( "Write a function to generate the n-th sequence values (bottle payment) for each Mike and Tim. Explain why you chose that function. <BR>\n" );
document.write( "Using the formula how much will each receive for the 50th bottle? For the 100th bottle? <BR>\n" );
document.write( "Determine the type of sequences for both Mike and Tim. <BR>\n" );
document.write( "Who do you think fairs better in the long term? Explain </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713619 by </B><A HREF=/tutors/aboutme.mpl?userid=jorel1380><B>jorel1380(3719)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jorel1380><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=jorel1380><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=713619\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let n be the number of bottles each picks up. Then:<BR>\n" );
document.write( "Mike's bottles: .1 + 0.1*(1.02)^(n-1) represents the total pay for n bottles he picks up (for n>1)<BR>\n" );
document.write( "Tim's bottles: (0.15+.01(n-1)) for n bottles that Tim picks up<BR>\n" );
document.write( "For Mike, the 50th bottle equals: .1+0.1*(1.02)^49=.1+.26=0.36<BR>\n" );
document.write( "For Tim, the 50th bottle is worth: .15+(.01)49=0.64<BR>\n" );
document.write( "For Mike, the term is geometric, because the price per bottle rises exponentially, while for Tim the price per bottle rises algebraically, by a constan)t 1 cent per extra bottle.  In the long run, Mike is going to make more per bottle.<BR>\n" );
document.write( "&#9786;&#9786;&#9786;&#9786; </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );