document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=825369>Question 825369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first and last term of a geometric series are 2 and 2048 respectively. The sum of the series is 273. Find the common ratio and the number of terms.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #497354 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=497354\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem is not possible.<br><BR>\n" );
document.write( "Here's why. The formula for the nth term of a geometric sequence is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28%28n-1%29%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > being the first term, \"r\" being the common ratio, and \"n\" being the number of the term. So 2 and 2048 should fit this formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2048+=+2%2Ar%5E%28%28n-1%29%29 ALIGN=MIDDLE  ALT=\"2048+=+2%2Ar%5E%28%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we realize that 2048 = 2*2*2*2*2*2*2*2*2*2*2 then we know that the only factors of 2048 are 2's or powers of 2. 2's and powers of 2 cannot add up to an odd number like 273.<br><BR>\n" );
document.write( "If we don't know this about 2048 we could try to go further. Dividing both sides by 2 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024+=+r%5E%28%28n-1%29%29 ALIGN=MIDDLE  ALT=\"1024+=+r%5E%28%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we will use the formula for the sum of a geometric series:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D+=+a%5B1%5D%28%281-r%5En%29%2F%281-r%29%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D+=+a%5B1%5D%28%281-r%5En%29%2F%281-r%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the sum of the first n terms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term, \"r\" is the common ratio, and \"n\" is the number of terms. Substituting in our values for the sum and the first term we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273+=+2%28%281-r%5En%29%2F%281-r%29%29 ALIGN=MIDDLE  ALT=\"273+=+2%28%281-r%5En%29%2F%281-r%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying each side by (1-r) [to eliminate the fraction] we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273%281-r%29+=+2%281-r%5En%29 ALIGN=MIDDLE  ALT=\"273%281-r%29+=+2%281-r%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273-273r+=+2%281-r%5En%29 ALIGN=MIDDLE  ALT=\"273-273r+=+2%281-r%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we return to our earlier equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024+=+r%5E%28%28n-1%29%29 ALIGN=MIDDLE  ALT=\"1024+=+r%5E%28%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we multiply each side of this by r we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar ALIGN=MIDDLE  ALT=\"1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the right side we use the rule for exponents when multiplying (i.e. add the exponents:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar%5E1 ALIGN=MIDDLE  ALT=\"1024%2Ar+=+r%5E%28%28n-1%29%29+%2Ar%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024r+=+r%5E%28%28n-1%29%2B1%29 ALIGN=MIDDLE  ALT=\"1024r+=+r%5E%28%28n-1%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024r+=+r%5En ALIGN=MIDDLE  ALT=\"1024r+=+r%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This gives us an expression to use back in:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273-273r+=+2%281-r%5En%29 ALIGN=MIDDLE  ALT=\"273-273r+=+2%281-r%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting in 1024r:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273-273r+=+2%281-1024r%29 ALIGN=MIDDLE  ALT=\"273-273r+=+2%281-1024r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can solve for r. Multiplying out the right side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273-273r+=+2-2048r ALIGN=MIDDLE  ALT=\"273-273r+=+2-2048r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding 2048r:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=273%2B1775r+=+2 ALIGN=MIDDLE  ALT=\"273%2B1775r+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 273:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1775r+=+-271 ALIGN=MIDDLE  ALT=\"1775r+=+-271\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing by 1775:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-271%29%2F1775 ALIGN=MIDDLE  ALT=\"r+=+%28-271%29%2F1775\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With this value for \"r\", it will be impossible for the series to start at 2 and eventually get to 2048. (Try it and see!).<br><BR>\n" );
document.write( "Since the formulas for geometric series do not work with a first term of 2, a last term of 2048 and a sum of 273, these numbers cannot reflect a geometric series. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );