document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102223>Question 1102223</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the sum of all natural numbers between 1 and 200 which are multiples of 5? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #716896 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=716896\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "This will be pretty much the same answer as provided by the other tutor, but using informal logical analysis instead of formal mathematical formulas.<br><BR>\n" );
document.write( "Assuming that \"between 1 and 200\" includes 200, the sum is 5+10+...+195+200.<br><BR>\n" );
document.write( "The number of terms in the sum is 200/5 = 40.<br><BR>\n" );
document.write( "Since the numbers are equally spaced, the 40 numbers can be paired into 20 pairs, each with a sum of 205: 5+200 = 205; 10+195 = 205; 15+190 = 205; etc.<br><BR>\n" );
document.write( "So the sum of all the numbers is the sum of 20 pairs, with each pair having a sum of 205: 205*20 = 4100.<br><BR>\n" );
document.write( "And here is an alternative way to think about getting the final sum, once we have found that there are 40 terms in the sum.<br><BR>\n" );
document.write( "Again since the numbers are equally spaced, the average of all the terms is the average of the first and last terms: (5+200)/2 = 102.5<br><BR>\n" );
document.write( "Then the sum of all the numbers is that average, multiplied by the number of terms: 40(102.5) = 4100.<br><BR>\n" );
document.write( "To summarize these two different ways of finding the sum, we have either<br><BR>\n" );
document.write( "(a) (number of pairs)*(sum of first and last) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2F2%29%28a1%2Ban%29 ALIGN=MIDDLE  ALT=\"%28n%2F2%29%28a1%2Ban%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "(b) (number of numbers)*(average of first and last) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28%28a1%2Ban%29%2F2%29 ALIGN=MIDDLE  ALT=\"n%28%28a1%2Ban%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Those two formulas, representing different ways of thinking of getting the answer, are clearly mathematically equivalent. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );