document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199566>Question 1199566</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The numbers 2, b, c, d, 72 are listed in increasing order so that 2, b, c form an arithmetic sequence, b, c, d form a geometric sequence, and c, d, 72 form a harmonic sequence. What is the value of b + c? </SPAN>\n" );
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document.write( "Let x be the common difference in the arithmetic sequence.  Then....<br><BR>\n" );
document.write( "2, b, and c form an arithmetic sequence, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+2%2Bx ALIGN=MIDDLE  ALT=\"b+=+2%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+2%2Bb+=+2%2B2x ALIGN=MIDDLE  ALT=\"c+=+2%2Bb+=+2%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "b, c, and d form a geometric sequence, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bd+=+c%5E2 ALIGN=MIDDLE  ALT=\"bd+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+c%5E2%2Fb ALIGN=MIDDLE  ALT=\"d+=+c%5E2%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "c, d, and e form a harmonic sequence, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fd=1%2Fc%2B1%2Fe ALIGN=MIDDLE  ALT=\"2%2Fd=1%2Fc%2B1%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fd-1%2Fc=1%2Fe ALIGN=MIDDLE  ALT=\"2%2Fd-1%2Fc=1%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282c-d%29%2Fcd=1%2Fe ALIGN=MIDDLE  ALT=\"%282c-d%29%2Fcd=1%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=cd%2F%282c-d%29 ALIGN=MIDDLE  ALT=\"e=cd%2F%282c-d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=x%2B2 ALIGN=MIDDLE  ALT=\"b=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2x%2B2 ALIGN=MIDDLE  ALT=\"c=2x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=c%5E2%2Fb=%282x%2B2%29%5E2%2F%28x%2B2%29 ALIGN=MIDDLE  ALT=\"d=c%5E2%2Fb=%282x%2B2%29%5E2%2F%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=cd%2F%282c-d%29=%28%282x%2B2%29%5E3%2F%28x%2B2%29%29%2F%282%282x%2B2%29-%28%282x%2B2%29%5E2%29%2F%28x%2B2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We need to have e = 72.  Simplify the expression for e, starting by multiplying numerator and denominator of the fraction by (x+2).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=%28%282x%2B2%29%5E3%29%2F%28%282%282x%2B2%29%28x%2B2%29-%282x%2B2%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"e=%28%282x%2B2%29%5E3%29%2F%28%282%282x%2B2%29%28x%2B2%29-%282x%2B2%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=%28%282x%2B2%29%5E3%29%2F%28%282x%2B2%29%282%28x%2B2%29-%282x%2B2%29%29%29 ALIGN=MIDDLE  ALT=\"e=%28%282x%2B2%29%5E3%29%2F%28%282x%2B2%29%282%28x%2B2%29-%282x%2B2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=%282x%2B2%29%5E3%2F%28%282x%2B2%29%282%29%29 ALIGN=MIDDLE  ALT=\"e=%282x%2B2%29%5E3%2F%28%282x%2B2%29%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=%282x%2B2%29%5E2%2F2+=+4%28x%2B1%29%5E2%2F2+=+2%28x%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"e=%282x%2B2%29%5E2%2F2+=+4%28x%2B1%29%5E2%2F2+=+2%28x%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then solve for e=72.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2B1%29%5E2=72 ALIGN=MIDDLE  ALT=\"2%28x%2B1%29%5E2=72\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2=36 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=6 ALIGN=MIDDLE  ALT=\"x%2B1=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And we have for the sequence of numbers:<BR>\n" );
document.write( "2<BR>\n" );
document.write( "b = 2+5 = 7<BR>\n" );
document.write( "c = 7+5 = 12<BR>\n" );
document.write( "d = 12^2/7 = 144/7<BR>\n" );
document.write( "e = (12(144/7))/(24-144/7) = (1728/7)/(24/7) = 72<br><BR>\n" );
document.write( "ANSWER: b+c = 7+12 = 19<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );