2508 /www/nrich/html/content/id/2508/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <div>The diagram shows that $1 + 3 + 5 + 7 + 5 + 3 + 1 = 3^2 + 4^2$.</div> <br></br> <div>What is $1 + 3 + 5 + \dots + 1999 + 2001 + 1999 + \dots + 5 + 3 + 1$?</div> <br></br> <div style="text-align: left;"><mdo:image alt="" height="88" src="Dots%202001%2013.gif" width="272"></mdo:image></div> <div style="text-align: left;"><br></br> If you liked this problem, <a href="http://nrich.maths.org/public/viewer.php?obj_id=2292&amp;refpage=titlesearch.php">here is an NRICH task</a> which challenges you to use similar mathematical ideas.  </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">The general form of the example given is $$1 + 3 + 5 + \dots + (2n - 1) + \dots + 5 + 3 + 1 = (n - 1)^2 + n^2$$<br></br> Therefore, for $n = 1001$: $$1 + 3 + 5 + \dots + 1999 + 2001 + 1999 + \dots + 5 + 3 + 1 = 1000^2 + 1001^2 = 2002001$$<br></br></mdoxml> 2 3 0 0 1 0 0 Weekly Problem 20-2013 Using, Applying and Reasoning about Mathematics Generalising Sequences, Functions and Graphs Series Admin Short problems