document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1064992>Question 1064992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Part (a): Find the sum a + (a + 1) + (a + 2) + ... + (a + n - 1) in terms of a and n. <BR>\n" );
document.write( "Part (b): Find all pairs of positive integers (a,n) such that n greater than 2 and a + (a + 1) + (a + 2) + ... + (a + n - 1) = 100.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #680126 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=680126\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Part (a):<BR>\n" );
document.write( "————<BR>\n" );
document.write( "a + (a+1) + (a+2) + … + (a+n-1)  <BR>\n" );
document.write( "can be re-written:<BR>\n" );
document.write( " (a+a+ … + a) + (1+2+ … + n-1)\r<BR>\n" );
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document.write( "There are  n a's in the first parenthesis, and the 2nd parenthesis sums to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%2A%28n-1%29+%2F+2+ ALIGN=MIDDLE  ALT=\"+n%2A%28n-1%29+%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "The sum S is:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28S+=+an+%2B+n%2A%28n-1%29%2F2%29+ ALIGN=MIDDLE  ALT=\"+highlight%28S+=+an+%2B+n%2A%28n-1%29%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "Part (b):<BR>\n" );
document.write( "————<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2An+%2B+n%2A%28n-1%29%2F2+=+100+ ALIGN=MIDDLE  ALT=\"+a%2An+%2B+n%2A%28n-1%29%2F2+=+100+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2an+%2B+n%28n-1%29+=+200+ ALIGN=MIDDLE  ALT=\"+2an+%2B+n%28n-1%29+=+200+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2an+%2B+n%5E2+-+n+=+200+ ALIGN=MIDDLE  ALT=\"+2an+%2B+n%5E2+-+n+=+200+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++n%5E2+%2B+2an+-+n+=+200+ ALIGN=MIDDLE  ALT=\"++n%5E2+%2B+2an+-+n+=+200+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++n%28n+%2B+2a+-+1%29+=+200+ ALIGN=MIDDLE  ALT=\"++n%28n+%2B+2a+-+1%29+=+200+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "a and n are integers, and both are > 0\r<BR>\n" );
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document.write( "We only need to try factors less than or equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28200%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28200%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  which is approx 14.14,  an taking the floor of this value gives an max integer value of 14 to try.<BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "Factor 200 to see which factors we should try:<BR>\n" );
document.write( "200 = 2*2*2*5*5<BR>\n" );
document.write( "—<BR>\n" );
document.write( "We  want to try all combinations of the factors given above, such that their product is <= 14.<BR>\n" );
document.write( "—<BR>\n" );
document.write( "Try n=2:    2(2+2a-1) = 200 —>   1+2a = 198   (won't work because 2a = 197  —> a is a non-integer)<BR>\n" );
document.write( "Try n=4:    4(4+2a-1) = 200 —>  3+2a = 196 —>  2a = 193  (no good, a is non-integer)<BR>\n" );
document.write( "Try n=8:    8(8+2a-1) = 200  —>  7+2a = 25 —> 2a = 18 —> a=9<BR>\n" );
document.write( "                               <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+a=9%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+a=9%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28n=8+%29 ALIGN=MIDDLE  ALT=\"+highlight%28n=8+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfy the equality\r<BR>\n" );
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document.write( "n=5:   5(5+2a-1) = 200  —>  4+2a = 40 —> 2a = 36 —> a = 18<BR>\n" );
document.write( "                  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28a=18%29+ ALIGN=MIDDLE  ALT=\"+highlight%28a=18%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28n=5%29+ ALIGN=MIDDLE  ALT=\"+highlight%28n=5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfy the equality\r<BR>\n" );
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document.write( "n=10:   10(10+2a-1) = 200 —>  9+2a = 20  (no good, a is non-integer)\r<BR>\n" );
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document.write( "n=20:  n is too large  \r<BR>\n" );
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document.write( "I think that is it. </SPAN>\n" );
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