document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202122>Question 1202122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On a string of pearls, the largest pearl is in the center and the smallest pearls are on the <BR>\n" );
document.write( "ends. Each of the small pearls on the two ends costs $1; each of the next larger pearls costs <BR>\n" );
document.write( "$2 each; the third pearl from each end costs $3 each; and so on. On the basis of this plan, <BR>\n" );
document.write( "how much would a string of 9 pearls cost? 12? 25? n? what equation can be used to find the cost of any number of pearls? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836776 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=836776\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The analysis of the problem is different depending on whether the number of pearls is even or odd.  Trying to find a single formula for the cost of any string of n pearls will be very awkward, so it's much better to find separate formulas for the two cases.<br><BR>\n" );
document.write( "First case: the number of pearls n is odd<br><BR>\n" );
document.write( "let the number of pearls be 2k+1, where k is an integer<br><BR>\n" );
document.write( "The cost of the first k+1 pearls is 1+2+...+k+(k+1); the cost of the last k pearls is k+(k-1)+...+2+1.  The total cost of the pearls is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28k%2B1%29%28k%2B2%29%29%2F2%2B%28k%28k%2B1%29%29%2F2=%28%28k%2B1%29%282k%2B2%29%2F2%29=%28k%2B1%29%28k%2B1%29=%28k%2B1%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So when the number of pearls is odd, n = 2k+1, the total cost of the pearls is (k+1)^2.<br><BR>\n" );
document.write( "Second case: the number of pearls is even<br><BR>\n" );
document.write( "let the number of pearls be 2k, where k is an integer<br><BR>\n" );
document.write( "The cost of the first k pearls is 1+2+...+k; and the sum of the last k pearls is the same.  The total cost of the pearls is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28k%28k%2B1%29%2F2%29=k%28k%2B1%29 ALIGN=MIDDLE  ALT=\"2%28k%28k%2B1%29%2F2%29=k%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWERS:<br><BR>\n" );
document.write( "If the number of pearls is odd (n=2k+1), then the total cost of the pearls is (k+1)^2<br><BR>\n" );
document.write( "If the number of pearls is even (n=2k), then the total cost of the pearls is k(k+1)<br><BR>\n" );
document.write( "Informally, determining the cost of a strand of n pearls can be described as follows:<br><BR>\n" );
document.write( "If n is odd, add 1 to n, divide by 2, and square the result.<BR>\n" );
document.write( "If n is even, divide n by 2 and multiply the result by the next integer.<br><BR>\n" );
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