document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1050433>Question 1050433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that if k is an integer then (4k+1)i^4k + (4k+2)i^4k+1 + (4k+3)i^4k+2 + (4k+4)i^4k+3= 2-2i. <BR>\n" );
document.write( "Use this to prove that <BR>\n" );
document.write( "1+2i+3i^2+4i^3+...+1995i^1994+1996i^1995=-998-998i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #666209 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666209\">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=i%5E%284k%29+=+%28i%5E4%29%5Ek+=+1%5Ek+=+1 ALIGN=MIDDLE  ALT=\"i%5E%284k%29+=+%28i%5E4%29%5Ek+=+1%5Ek+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "===><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E%284k%2B1%29+=+i ALIGN=MIDDLE  ALT=\"i%5E%284k%2B1%29+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E%284k%2B2%29+=+-1 ALIGN=MIDDLE  ALT=\"i%5E%284k%2B2%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E%284k%2B3%29+=+-i ALIGN=MIDDLE  ALT=\"i%5E%284k%2B3%29+=+-i\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284k%2B1%29i%5E%284k%29+%2B+%284k%2B2%29i%5E%284k%2B1%29+%2B+%284k%2B3%29i%5E%284k%2B2%29+%2B+%284k%2B4%29i%5E%284k%2B3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284k%2B1%29%2A1+%2B+%284k%2B2%29%2Ai+%2B+%284k%2B3%29%2A%28-1%29+%2B+%284k%2B4%29%2A%28-i%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+-+2i ALIGN=MIDDLE  ALT=\"-2+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now substitute the values of k = 0, 1, 2, 3,...,497, 498 into the formula.\r<BR>\n" );
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document.write( "On the left side of the equation, we would get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2i%2B3i%5E2%2B4i%5E3 ALIGN=MIDDLE  ALT=\"1%2B2i%2B3i%5E2%2B4i%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1995i%5E1994%2B1996i%5E1995 ALIGN=MIDDLE  ALT=\"1995i%5E1994%2B1996i%5E1995\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "while on the right side, we would get\r<BR>\n" );
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document.write( "499*(-2 - 2i) = -998 - 998i.\r<BR>\n" );
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document.write( "Therefore,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2i%2B3i%5E2%2B4i%5E3 ALIGN=MIDDLE  ALT=\"1%2B2i%2B3i%5E2%2B4i%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1995i%5E1994%2B1996i%5E1995+=+-998-998i ALIGN=MIDDLE  ALT=\"1995i%5E1994%2B1996i%5E1995+=+-998-998i\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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