document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=761962>Question 761962</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me on this problem:  Suppose \"k\" is some positive interger. Consider numbers of the form &#8730;n) where \"n\" is a positive interger. In terms of \"k,\" how many numbers of this form differ from \"k\" bh less than 1? Explain why your answer works in general. (Hint: for this question it may help to try some small values of \"k\" and search for a pattern in the results. For example, if k= 2 there are 7 numbers of the desired form that differ from \"k\" by less than one since &#8730;2), &#8730;3), &#8730;4), &#8730;5), &#8730;6), &#8730;7), and &#8730;8) are the only such numbers that differ from 2 by less than 1. Now, try k=3 and so forth. After you find a formula think about why your formula works in general.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #463615 by </B><A HREF=/tutors/aboutme.mpl?userid=ramkikk66><B>ramkikk66(644)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ramkikk66><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=463615\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For any number k,\r<BR>\n" );
document.write( "\n" );
document.write( "all values of n in the range (k-1)^2 and (k+1)^2 will satisfy the condition, where sqrt(n) would differ from k by less than 1.\r<BR>\n" );
document.write( "\n" );
document.write( "How many numbers are there between k^2 and (k+1)^2?\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B1%29%5E2+-+k%5E2+=+2%2Ak+%2B+1 ALIGN=MIDDLE  ALT=\"%28k%2B1%29%5E2+-+k%5E2+=+2%2Ak+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "How many numbers are there between k^2 and (k-1)^2?\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%29%5E2+-+%28k-1%29%5E2+=+k%5E2+-+%28k%5E2+-+2k+%2B+1%29+=+2%2Ak+-+1 ALIGN=MIDDLE  ALT=\"%28k%29%5E2+-+%28k-1%29%5E2+=+k%5E2+-+%28k%5E2+-+2k+%2B+1%29+=+2%2Ak+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Sum of the two = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ak ALIGN=MIDDLE  ALT=\"4%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "However, this list of 4k numbers also includes k twice. So we have to reduce it by 1.\r<BR>\n" );
document.write( "\n" );
document.write( "So, total possible values of n where sqrt(n) differs from k by less than 1,\r<BR>\n" );
document.write( "\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%284%2Ak+-+1%29 ALIGN=MIDDLE  ALT=\"highlight%284%2Ak+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( ":)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check a few samples:<BR>\n" );
document.write( "For k = 2, the set of n would be sqrt(2) to sqrt(8) = 7 numbers = 4*2 - 1<BR>\n" );
document.write( "For k = 3, the set of n would be sqrt(5) to sqrt(15) = 11 numbers = 4*3 - 1<BR>\n" );
document.write( "For k = 5, the set of n would be sqrt(17) to sqrt(35) = 19 numbers = 4*5 - 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );