document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209367>Question 1209367</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of all positive integers less than 1000 ending in 3 or 4 or 5 or 8. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848686 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848686\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This is a slight variation of what tutor greenestamps did.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3+4+5+8 = 20<BR>\n" );
document.write( "Add 10 to each item on the left side, so you're adding 40 to both sides<BR>\n" );
document.write( "13+14+15+18 = 60<BR>\n" );
document.write( "Repeat the previous step<BR>\n" );
document.write( "23+24+25+28 = 100<BR>\n" );
document.write( "Repeat again<BR>\n" );
document.write( "33+34+35+38 = 140<BR>\n" );
document.write( "And so on.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The final four terms would add to 993+994+995+998 = 3980\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The right hand sides of those equations form this arithmetic sequence 20, 60, 100, 140, ..., 3980\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first terms of each left hand side are 3,13,23,33,...,993. <BR>\n" );
document.write( "Erase the units digit to arrive at 0,1,2,3,...,99<BR>\n" );
document.write( "This shows we have 99-0+1 = 100 equations and therefore we have 100 terms in the sequence 20, 60, 100, 140, ..., 3980\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "S<sub>n</sub> = sum of first n terms of arithmetic sequence<BR>\n" );
document.write( "S<sub>n</sub> = (n/2)*(firstTerm + nthTerm)<BR>\n" );
document.write( "S<sub>100</sub> = (100/2)*(20 + 3980)<BR>\n" );
document.write( "S<sub>100</sub> = <font color=red>200,000</font> is the final answer.<BR>\n" );
document.write( "The comma separator is there to make the single number more readable.<BR>\n" );
document.write( "Erase the comma if needed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another way to arrive at this answer is to say,<BR>\n" );
document.write( "S<sub>n</sub> = (n/2)*(2*a + d*(n-1))<BR>\n" );
document.write( "S<sub>100</sub> = (100/2)*(2*20 + 40*(100-1))<BR>\n" );
document.write( "S<sub>100</sub> = <font color=red>200,000</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );