document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30794>Question 30794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question is really confusing, my tutor ended up answering it in class but everyone was still kinda dazed when we left class.  Will you please try to explain or solve this in a manner that is within a students grasp.\r<BR>\n" );
document.write( "\n" );
document.write( "Question:<BR>\n" );
document.write( "When the first n positive integers are added together, their sum is given by:<BR>\n" );
document.write( "1/2n(n+1)\r<BR>\n" );
document.write( "\n" );
document.write( "a) Demonstrate that this result holds for the case  n=5.<BR>\n" );
document.write( "b) Find the value of n for which the sum is 105.<BR>\n" );
document.write( "c) What is the smallest value of n for which the sum exceeds 1000?\r<BR>\n" );
document.write( "\n" );
document.write( "The chapter we are working on is based on factoring.  Please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #17517 by </B><A HREF=/tutors/aboutme.mpl?userid=Fermat><B>Fermat(136)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fermat><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=Fermat><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=17517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi there, I'n not too sure what exactly the confusion might be, so I'll just go through the problems and hope that helps. If you are still having problems, then you can relist this question, with additional points you would like to be addressed, or you can email me for info. I'm not absolutely sure how you can email me, but I thnk you can do that when you view this answer. (Look around the web page)<BR>\n" );
document.write( "Sn = (1/2)n(n+1)<BR>\n" );
document.write( "a) We have to show that the above formula holds for n=5.<BR>\n" );
document.write( "If n is 5, then we want to add up the 1st 5 natural numbers, like so<BR>\n" );
document.write( "S5 = 1 + 2 + 3 + 4 + 5<BR>\n" );
document.write( "S5 = 3 + 7 + 5 (adding two numbers at a time)<BR>\n" );
document.write( "S5 = 10 + 5<BR>\n" );
document.write( "S5 = 15<BR>\n" );
document.write( "=======<BR>\n" );
document.write( "If we now use the formula for Sn, with n = 5, then we get<BR>\n" );
document.write( "S5 = (1/2)5(6)<BR>\n" );
document.write( "S5 = (1/2)(30)<BR>\n" );
document.write( "S5 = 15<BR>\n" );
document.write( "=======<BR>\n" );
document.write( "So, the formula holds for n = 5.<BR>\n" );
document.write( "================================<BR>\n" );
document.write( "b) Find the value of n for which the sum is 105.<BR>\n" );
document.write( "If Sn = 105, the we can write out an equation,<BR>\n" );
document.write( "Sn = (1/2)n(n+1) = 105<BR>\n" );
document.write( "(1/2)n(n+1) = 105<BR>\n" );
document.write( "n(n+1) = 210 (multiplying both sides by 2)<BR>\n" );
document.write( "n² + n = 210 ( expanding the lhs)<BR>\n" );
document.write( "n² + n - 210 = 0 ( rearranging the eqn)<BR>\n" );
document.write( "(n + 15)(n - 14) = 0 (factoring the terms)<BR>\n" );
document.write( ".: n = -15, n = 14<BR>\n" );
document.write( "==================<BR>\n" );
document.write( "Although n = -15 satisfies the quadratic eqn, it is not a valid solution since it is negative and hence not a natural number.<BR>\n" );
document.write( "Ans: n = 14<BR>\n" );
document.write( "===========<BR>\n" );
document.write( "c) What is the smallest value of n for which the sum exceeds 1000?<BR>\n" );
document.write( "This is similar to part b), but is an inequality.<BR>\n" );
document.write( "This time we can write,<BR>\n" );
document.write( "Sn > 1000<BR>\n" );
document.write( "substituting the formula for Sn,<BR>\n" );
document.write( "(1/2)n(n+1) > 1000<BR>\n" );
document.write( "n(n+1) > 2000<BR>\n" );
document.write( "n² + n > 2000<BR>\n" );
document.write( "Now you can go all mathematical here and make up a quadratic eqn, viz n² + n - 2000 = 0 and then use the quadratic formula to find roots of this equation and then test those roots in the original inequality to find out which one works,<BR>\n" );
document.write( "or,<BR>\n" );
document.write( "you can take a guess!<BR>\n" );
document.write( "n² is bigger than n, much bigger if n is any decent size. So for a rough approximation, let<BR>\n" );
document.write( "n² > 2000<BR>\n" );
document.write( "So what's the square root of 2000?<BR>\n" );
document.write( "Using my trusty calculator, I get 44.72<BR>\n" );
document.write( "Since you have to add on n to the n² bit then try, as a first guess, n=44.<BR>\n" );
document.write( "44² + 44 = 1980 (too small, so try the next number up)<BR>\n" );
document.write( "45² + 45 = 2070 - bingo!<BR>\n" );
document.write( "Ans: n= 45 is smallest number such that Sn > 1000<BR>\n" );
document.write( "=================================================<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );