document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091859>Question 1091859</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help...<BR>\n" );
document.write( "I know the arithmetic sequence Sn=n(A1+An)/2, but I cannot solve this problem<BR>\n" );
document.write( "Q: nth triangular number = n(n+1)/2, <BR>\n" );
document.write( "Prove this formula algebraically using the sum of a finite arithmetic sequence theorem. <BR>\n" );
document.write( "Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #706343 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=706343\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The Nth Triangular number is the sum of the sequence of positive integers up to and including N:<BR>\n" );
document.write( "Noting that  A1 = 1  and An = n:\r<BR>\n" );
document.write( "\n" );
document.write( "Tn = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sum%28k%2C+k=1%2C+n%29+=+%281%2F2%29n%28A1+%2B+An%29+=+%281%2F2%29n%28n%2B1%29+ ALIGN=MIDDLE  ALT=\"+sum%28k%2C+k=1%2C+n%29+=+%281%2F2%29n%28A1+%2B+An%29+=+%281%2F2%29n%28n%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "(you are just picking out one Tn, not the sum of an entire sequence of triangular numbers)\r<BR>\n" );
document.write( "\n" );
document.write( "—<BR>\n" );
document.write( "That concludes the proof IF you are allowed to use the starting point  Sn=n(A1+An)/2.\r<BR>\n" );
document.write( "\n" );
document.write( "———<BR>\n" );
document.write( "Here is more of a ground-up method that also proves Tn = n(n+1)/2 and then Sn=n(A1+An)/2:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B1%5D+=+1+ ALIGN=MIDDLE  ALT=\"+S%5B1%5D+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >                  — also the 1st triangular number<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B2%5D+=+1%2B2+=+3+ ALIGN=MIDDLE  ALT=\"+S%5B2%5D+=+1%2B2+=+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >      — also the 2nd triangular number<BR>\n" );
document.write( ": :                :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5Bn%5D+=+1%2B2+ ALIGN=MIDDLE  ALT=\"+S%5Bn%5D+=+1%2B2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > + … + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B%28n-1%29+%2B+n+ ALIGN=MIDDLE  ALT=\"%2B%28n-1%29+%2B+n+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  — also the  n'th triangular number<BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5Bn%5D+ ALIGN=MIDDLE  ALT=\"+S%5Bn%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > you can take the largest number and pair it with the smallest (1) and form n+1, then (n-2) and 3 to get n+1, etc.  You can do this  n/2 times*.  Thus the sum 1+2+…+n  is  n(n+1)/2.<BR>\n" );
document.write( "—<BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "* To see the \"n/2 times\" because it may not be obvious (also n may be odd) just do this:  form the pairs to be summed as (1,n), (2,n-1), (3,n-2), … ,  (n-2,3), (n-1,2), (n,1)  and note that we show each pair TWICE.\r<BR>\n" );
document.write( "\n" );
document.write( "The first number in each pair counts how many pairs you have:  n<BR>\n" );
document.write( "The sum of each pair is:  n+1<BR>\n" );
document.write( "The sum of all the pairs shown:   n*(n+1)<BR>\n" );
document.write( "Divide by 2 because we double-counted:    Sn = n*(n+1)/2   (done)<BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "Proof of  Sn = n(A1+An)/2:\r<BR>\n" );
document.write( "\n" );
document.write( "    Sn = A1+A2+…+An\r<BR>\n" );
document.write( "\n" );
document.write( "Pair them up as we did above:  (A1,An), (A2,An-1), … ,(An-1,A2), (An,A1)\r<BR>\n" );
document.write( "\n" );
document.write( "Thats n pairs.<BR>\n" );
document.write( "Sum of each pair is  A1+An   (A2 + An-1 = A1+An because A2 = A1 + d and An-1 = An - d,  similar logic for other pairs)<BR>\n" );
document.write( "Adjust for double counting:  Sum =  n*(A1+An)/2   \r<BR>\n" );
document.write( "\n" );
document.write( "—<BR>\n" );
document.write( "There are other ways to do the proofs, this is just the way I think about it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );