document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1116978>Question 1116978</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine all nonnegative integers r such that it is possible for an infinite geometric sequence to contain exactly r terms that are integers. Prove your answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #731930 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=731930\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I do not believe ther is such a nonnegative integer, but if you know of a different answer, please enlighten me.\r<BR>\n" );
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document.write( "A nonnegative integer is either a positive integer or it is zero.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a positive integer,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer term of an infinite geometric sequence with common ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%2B1%5D=a%5Bn%5D%2Ar ALIGN=MIDDLE  ALT=\"a%5Bn%2B1%5D=a%5Bn%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be an integer,<BR>\n" );
document.write( "and so will be every term after that.<BR>\n" );
document.write( "As a consequence, there will be infinite terms that are integers.<BR>\n" );
document.write( "In that case, it will not be possible for that infinite geometric sequence to contain exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms that are integers.<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=0 ALIGN=MIDDLE  ALT=\"r=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , regardless of the value of first term <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;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=a%5B1%5D%2A0=0 ALIGN=MIDDLE  ALT=\"a%5B2%5D=a%5B1%5D%2A0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and all subsequent terms will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
document.write( "In that case, there will also be an infinite number of terms that are integers,<BR>\n" );
document.write( "and that infinite geometric sequence will not contain exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms that are integers either. </SPAN>\n" );
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