{"id":1511,"date":"2021-04-22T02:10:10","date_gmt":"2021-04-22T02:10:10","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=1511"},"modified":"2024-10-25T02:46:32","modified_gmt":"2024-10-25T06:46:32","slug":"2020-mathcounts-state-sprint-round-30","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=1511","title":{"rendered":"2020 Mathcounts State Sprint Round #30"},"content":{"rendered":"<p>Hank builds an increasing sequence of positive integers as follows: The first term is 1 and the second term is 2. Each subsequent term is the smallest positive integer that does NOT form a three-term arithmetic sequence with any previous terms of the sequence. The first five terms of Hank\u2019s sequence are 1, 2, 4, 5, 10. How many of the first 729 positive integers are terms in Hank\u2019s sequence? Click <a onclick=\"toggle_visibility('2020-mathcounts-state-30');\">here<\/a> for the solution.<\/p>\n<div id=\"2020-mathcounts-state-30\" style=\"display:none\">\n<p><strong>Solution<\/strong>: It is important to note that $729=3^6$. If we take a look at the smaller powers of 3, namely $3^i, i = 0,1,2,3$, we might be able to find a pattern. With $3^0$, it is obvious: 1 is the only number that exists in the sequence, and for $3^1$ there are two: 1, 2. For $3^2$, the problem lists them for us: 1,2,4,5, which is 4. We can notice a pattern here: For $3^0$ it is 1, for $3^1$ it is 2, for $3^2$ it is 4, so let\u2019s check for $3^3$. Continuing the sequence, we get<br \/>\n$$1,2,4,5,10,11,13,14$$<br \/>\nThere are 8 terms here, so we definitely have a pattern. For each power of 3, the number of sequence elements is two to the power of the same number. So for the first 729 terms, the number of terms is $2^6=\\boxed{64}$.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hank builds an increasing sequence of positive integers as follows: The first term is 1 and the second term is 2. Each subsequent term is the smallest positive integer that does NOT form a three-term arithmetic sequence with any previous &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=1511\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[5,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1511"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1511"}],"version-history":[{"count":4,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1511\/revisions"}],"predecessor-version":[{"id":4570,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/1511\/revisions\/4570"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}