document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460555>Question 460555</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The average of the squares of seven consecutives integers is 53. The average of these integers  is? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #315927 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315927\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x-3, x-2, ..., x+3 be the seven integers, with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{(x-3)^2 + (x-2)^2 + ... + (x+3)^2}{7} = 53\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x-3)^2 + (x-2)^2 + ... + (x+3)^2 = 371\">\r<BR>\n" );
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document.write( "If we were to expand this all out, we would see that all the x terms would disappear (-6x on the left term cancels out 6x on the right, and so on). We are left with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x^2 + 9) + (x^2 + 4) + (x^2 + 1) + x^2 + (x^2 + 1) + (x^2 + 4) + (x^2 + 9) = 371\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 7x^2 + 28 = 371\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 7x^2 = 343\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 = 49 \Rightarrow x = \pm 7\">.\r<BR>\n" );
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document.write( "The average of the integers x-3, x-2, ..., x+3 is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{(x-3) + (x-2) + ... + (x+3)}{7}\"> Again, the constant terms cancel, leaving\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{7x}{7} = x\">\r<BR>\n" );
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document.write( "So the average of the original seven integers is either 7 or -7. </SPAN>\n" );
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