{"id":299,"date":"2020-02-22T03:14:22","date_gmt":"2020-02-22T03:14:22","guid":{"rendered":"http:\/\/blog.newtonchineseschool.org\/lilijia\/?p=299"},"modified":"2020-02-22T03:14:22","modified_gmt":"2020-02-22T03:14:22","slug":"sums-of-consecutive-counting-numbers","status":"publish","type":"post","link":"https:\/\/blog.newtonchineseschool.org\/lilijia\/2020\/02\/22\/sums-of-consecutive-counting-numbers\/","title":{"rendered":"Sums of Consecutive Counting Numbers"},"content":{"rendered":"\n<p><strong>Sums of Consecutive Counting Numbers<\/strong><\/p>\n\n\n\n<p>by Mr. John Bookston<\/p>\n\n\n\n<p>Definition:&nbsp; A sum\nof consecutive counting numbers can start with any counting number add at least\nthe next bigger counting number and stop adding after any number of consecutive\nintegers.<\/p>\n\n\n\n<p><strong>Examples:<\/strong>&nbsp; 3+4 = 7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7 is a consecutive sum of\ncounting numbers<\/p>\n\n\n\n<p>&nbsp;&nbsp; 1+2+3+4+5 =\n15&nbsp;&nbsp; &nbsp;4+5+6 = 15 and 7+8 = 15&nbsp;&nbsp;&nbsp;&nbsp; 15 is a consecutive sum in 3 distinct\nways.<\/p>\n\n\n\n<p>Investigation:&nbsp; <\/p>\n\n\n\n<p>For numbers that are consecutive sums, investigate patterns to predict the number of distinct ways larger numbers can be written as consecutive sums. \u00a0<\/p>\n\n\n\n<p><strong>Starters:<\/strong>&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1\nand 2 are not consecutive sums<\/p>\n\n\n\n<p>Every\n<strong>odd<\/strong> integer starting with 3 can be\nwritten as a consecutive sum of two counting numbers:<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(n-1)\/2 &nbsp;+ &nbsp;(n+1)\/2&nbsp; = &nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for every odd number, n.<\/p>\n\n\n\n<p>Make\na conjecture about which numbers are expressible as \u201cthe sum of 3 consecutive\ncounting numbers\u201d.&nbsp; <\/p>\n\n\n\n<p>Extension:\nDo the same for numbers that can be expressed as sums of 4,5 and 6 consecutive\ncounting numbers.<\/p>\n\n\n\n<p>Extension:&nbsp; Consider the prime factorization of the\ncounting numbers that cannot be expressed as a consecutive sum.&nbsp; Make a conjecture as to which counting\nnumbers less than 1000 fall into that category.<\/p>\n\n\n\n<p>Enjoy. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n\n\n\n<p>Good luck.\u00a0 <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sums of Consecutive Counting Numbers by Mr. John Bookston Definition:&nbsp; A sum of consecutive counting numbers can start with any counting number add at least the next bigger counting number and stop adding after any number of consecutive integers. Examples:&nbsp; 3+4 = 7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7 is a consecutive sum of counting numbers &nbsp;&nbsp; 1+2+3+4+5 = 15&nbsp;&nbsp; [&hellip;]<\/p>\n","protected":false},"author":157,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0},"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/299"}],"collection":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/users\/157"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/comments?post=299"}],"version-history":[{"count":1,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/299\/revisions"}],"predecessor-version":[{"id":300,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/299\/revisions\/300"}],"wp:attachment":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/media?parent=299"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/categories?post=299"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/tags?post=299"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}