document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=568043>Question 568043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please explain how to find only thee sum of the odd numbers in sigma notation n=1-35 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #366848 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=366848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You could say that odd numbers can be written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n-1 ALIGN=MIDDLE  ALT=\"2n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "All odd numbers can be written that way for some natural number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n-1 ALIGN=MIDDLE  ALT=\"2n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you an odd number for every natural number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A1-1=1 ALIGN=MIDDLE  ALT=\"2%2A1-1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2-1=3 ALIGN=MIDDLE  ALT=\"2%2A2-1=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ... <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A18-1=35 ALIGN=MIDDLE  ALT=\"2%2A18-1=35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You are trying to find the sum of the first 18 odd numbers:<BR>\n" );
document.write( "S=1+3+5+ ... +31+33+35<BR>\n" );
document.write( "There are many ways to go about it.<BR>\n" );
document.write( "BLAZING YOUR OWN TRAIL (or rediscovering gunpowder):<BR>\n" );
document.write( "2S=S+S=(1+3+5+ ... +31+33+35)+(1+3+5+ ... +31+33+35)=1+3+5+ ... +31+33+35+1+3+5+ ... +31+33+35.<BR>\n" );
document.write( "Because of commutative and associative properties, we can re-arrange that sum, pairing the first number with the last, second with second last, and so on, to get:<BR>\n" );
document.write( "2S=(1+35)+(3+33)+(5+31)+ ... +(31+5)+(33+3)+(35+1)=36+36+36+ ... +36+36+36=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2A36 ALIGN=MIDDLE  ALT=\"18%2A36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2S=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2A36 ALIGN=MIDDLE  ALT=\"18%2A36\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> S=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2A36%2F2 ALIGN=MIDDLE  ALT=\"18%2A36%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=324 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28S=324%29 ALIGN=MIDDLE  ALT=\"highlight%28S=324%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "INVOKING ARITHMETIC SEQUENCES (if that's what you are studying)<BR>\n" );
document.write( "The sum that you want is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%282n-1%2C1%2C18%29 ALIGN=MIDDLE  ALT=\"sum%282n-1%2C1%2C18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the sum of an arithmetic sequence (or arithmetic progression). You may write the terms explicit formula as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=1%2B2%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=1%2B2%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=1 ALIGN=MIDDLE  ALT=\"a%5B1%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and common difference <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=2 ALIGN=MIDDLE  ALT=\"d=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "(That is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=1%2B2n-2=2n-1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=1%2B2n-2=2n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "Naming conventions may vary, and you may use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B0%5D=-1 ALIGN=MIDDLE  ALT=\"a%5B0%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You may have been given the formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28n%2F2%29%282a%5B1%5D%2B%28n-1%29%2Ad%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28n%2F2%29%282a%5B1%5D%2B%28n-1%29%2Ad%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You would calculate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B18%5D=%2818%2F2%29%282%2A1%2B%2818-1%29%2A2%29=9%2A%282%2B17%2A2%29=9%2A%282%2B34%29=9%2A36=324 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "knowing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=35 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=35\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the start, you may rather calculate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%2818%2F2%29%281%2B35%29=9%2A36=324 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%2818%2F2%29%281%2B35%29=9%2A36=324\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Or maybe your naming conventions and formulas are a bit different.<BR>\n" );
document.write( "Anyway, you'll get the answer  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28S=324%29 ALIGN=MIDDLE  ALT=\"highlight%28S=324%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "INVOKING KNOWN SUMS<BR>\n" );
document.write( "You may have been told that the sum of the fist n natural numbers is<BR>\n" );
document.write( "1+2+3+ ...+(n-1)+n=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, the sum of all integers from 1 to 35 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2835%2B1%29%2F2=35%2A36%2F2=630 ALIGN=MIDDLE  ALT=\"35%2835%2B1%29%2F2=35%2A36%2F2=630\"   DISABLED_style= \"max-width:90%; height:auto;\" > and can be written as the sum of the odd numbers from 1 to 25 plus the sum of the first 17 even numbers from 2 to 34:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2C1%2C35%29=630=sum%282n-1%2C1%2C18%29%2Bsum%282k%2C1%2C17%29 ALIGN=MIDDLE  ALT=\"sum%28i%2C1%2C35%29=630=sum%282n-1%2C1%2C18%29%2Bsum%282k%2C1%2C17%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sum of the first 17 even numbers is twice the sum of the first 17 natural numbers:<BR>\n" );
document.write( "(2+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%282k%2C1%2C18%29=2sum%28k%2C1%2C18%29=2%2817%2817%2B1%29%2F2%29=2%2817%2A18%2F2%29=17%2A18=306 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=630=sum%282n-1%2C1%2C18%29%2B306 ALIGN=MIDDLE  ALT=\"630=sum%282n-1%2C1%2C18%29%2B306\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%282n-1%2C1%2C18%29=630-306=324 ALIGN=MIDDLE  ALT=\"sum%282n-1%2C1%2C18%29=630-306=324\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Anyway, you'll get the answer  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28S=324%29 ALIGN=MIDDLE  ALT=\"highlight%28S=324%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );