document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=696783>Question 696783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> check the formula \r<BR>\n" );
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document.write( "&#8721;[n=0,k,n^2]=k(k+1)(2k+1)/(6)<BR>\n" );
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document.write( "is corrrect for k = 1,2 and 3\r<BR>\n" );
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document.write( "( Fact: the formula works for all positive integers k) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #429367 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=429367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28n%5E2%2C+n=0%2C+k%29=+k%28k%2B1%29%282k%2B1%29%2F6 ALIGN=MIDDLE  ALT=\"sum%28n%5E2%2C+n=0%2C+k%29=+k%28k%2B1%29%282k%2B1%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For k=1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28n%5E2%2C+n=0%2C+k%29=sum%28n%5E2%2C+n=0%2C+1%29=+0%5E2%2B1%5E2=0%2B1=1 ALIGN=MIDDLE  ALT=\"sum%28n%5E2%2C+n=0%2C+k%29=sum%28n%5E2%2C+n=0%2C+1%29=+0%5E2%2B1%5E2=0%2B1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%28k%2B1%29%282k%2B1%29%2F6=1%2A%281%2B1%29%2A%282%2A1%2B1%29%2F6=1%2A2%2A3%2F6=6%2F6=1 ALIGN=MIDDLE  ALT=\"k%28k%2B1%29%282k%2B1%29%2F6=1%2A%281%2B1%29%2A%282%2A1%2B1%29%2F6=1%2A2%2A3%2F6=6%2F6=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the formula is valid for k=1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For k=2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28n%5E2%2C+n=0%2C+k%29=+sum%28n%5E2%2C+n=0%2C+2%29=+0%5E2%2B1%5E2%2B2%5E2=0%2B1%2B4=5 ALIGN=MIDDLE  ALT=\"sum%28n%5E2%2C+n=0%2C+k%29=+sum%28n%5E2%2C+n=0%2C+2%29=+0%5E2%2B1%5E2%2B2%5E2=0%2B1%2B4=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%28k%2B1%29%282k%2B1%29%2F6=2%2A%282%2B1%29%2A%282%2A2%2B1%29%2F6=2%2A3%2A5%2F6=30%2F6=5 ALIGN=MIDDLE  ALT=\"k%28k%2B1%29%282k%2B1%29%2F6=2%2A%282%2B1%29%2A%282%2A2%2B1%29%2F6=2%2A3%2A5%2F6=30%2F6=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the formula is valid for k=2.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For k=3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28n%5E2%2C+n=0%2C+k%29=+sum%28n%5E2%2C+n=0%2C+3%29=+0%5E2%2B1%5E2%2B2%5E2%2B3%5E2=0%2B1%2B4%2B9=14 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%28k%2B1%29%282k%2B1%29%2F6=3%2A%283%2B1%29%2A%282%2A3%2B1%29%2F6=3%2A4%2A7%2F6=84%2F6=14 ALIGN=MIDDLE  ALT=\"k%28k%2B1%29%282k%2B1%29%2F6=3%2A%283%2B1%29%2A%282%2A3%2B1%29%2F6=3%2A4%2A7%2F6=84%2F6=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the formula is valid for k=3.<BR>\n" );
document.write( " <BR>\n" );
document.write( "BIG SUM COMPUTATION (step by step):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%282m%5E2%2B3m-4%29%2C+m=0%2C+5%29=sum%282m%5E2%2C+m=0%2C+5%29%2Bsum%283m%2C+m=0%2C+5%29%2Bsum%28-4%2C+m=0%2C+5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Changing the order and grouping in the sum, just like you do when you add polynomials and \"collect like terms\").<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%282m%5E2%2B3m-4%29%2C+m=0%2C+5%29=sum%282m%5E2%2C+m=0%2C+5%29%2Bsum%283m%2C+m=0%2C+5%29%2Bsum%28-4%2C+m=0%2C+5%29=2%2Asum%28m%5E2%2C+m=0%2C+5%29%2B3%2Asum%28m%2C+m=0%2C+5%29-4%2Asum%281%2C+m=0%2C+5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(taking out common factors as the next step)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%282m%5E2%2B3m-4%29%2C+m=0%2C+5%29=sum%282m%5E2%2C+m=0%2C+5%29%2Bsum%283m%2C+m=0%2C+5%29%2Bsum%28-4%2C+m=0%2C+5%29=2%2Asum%28m%5E2%2C+m=0%2C+5%29%2B3%2Asum%28m%2C+m=0%2C+5%29-4%2Asum%281%2C+m=0%2C+5%29=5%285%2B1%29%282%2A5%2B1%29%2F6%2B3%2A15-4%2A6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(applying the formula given and easy to calculate values for the second and third sums).<BR>\n" );
document.write( "Now we finish the indicated calculations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%282m%5E2%2B3m-4%29%2C+m=0%2C+5%29=sum%282m%5E2%2C+m=0%2C+5%29%2Bsum%283m%2C+m=0%2C+5%29%2Bsum%28-4%2C+m=0%2C+5%29=2%2Asum%28m%5E2%2C+m=0%2C+5%29%2B3%2Asum%28m%2C+m=0%2C+5%29-4%2Asum%281%2C+m=0%2C+5%29=2%2A%285%285%2B1%29%282%2A5%2B1%29%2F6%29%2B3%2A15-4%2A6=2%2A55%2B45-24=110%2B45-24=highlight%28131%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "KNOWN SUMS:<BR>\n" );
document.write( "You know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%281%2C+m=0%2C+5%29=1%2B1%2B1%2B1%2B1%2B1=6 ALIGN=MIDDLE  ALT=\"sum%281%2C+m=0%2C+5%29=1%2B1%2B1%2B1%2B1%2B1=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > because it is a sum of six terms (m=0 to m=5) and all the terms are 1.<BR>\n" );
document.write( "The calculation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28m%2C+m=0%2C+5%29=0%2B1%2B2%2B3%2B4%2B5 ALIGN=MIDDLE  ALT=\"sum%28m%2C+m=0%2C+5%29=0%2B1%2B2%2B3%2B4%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also easy.<BR>\n" );
document.write( "You know that is an arithmetic series.<BR>\n" );
document.write( "You can calculate it by adding the terms I already listed above.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28m%2C+m=0%2C+5%29=0%2B1%2B2%2B3%2B4%2B5=15 ALIGN=MIDDLE  ALT=\"sum%28m%2C+m=0%2C+5%29=0%2B1%2B2%2B3%2B4%2B5=15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Otherwise you can apply the formula for sum of an arithmetic sequence<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28a%5Bm%5D%2C+m=0%2C+k%29=%28a%5B0%5D%2Ba%5Bk%5D%29%2A%28k%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28a%5Bm%5D%2C+m=0%2C+k%29=%28a%5B0%5D%2Ba%5Bk%5D%29%2A%28k%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28a%5Bm%5D%2C+m=0%2C+5%29=%280%2B5%29%2A%285%2B1%29%2F2=5%2A6%2F2=15 ALIGN=MIDDLE  ALT=\"sum%28a%5Bm%5D%2C+m=0%2C+5%29=%280%2B5%29%2A%285%2B1%29%2F2=5%2A6%2F2=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );