{"id":4120,"date":"2024-05-28T22:01:45","date_gmt":"2024-05-29T02:01:45","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4120"},"modified":"2025-03-09T22:39:24","modified_gmt":"2025-03-10T02:39:24","slug":"algebra-challenge-1","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4120","title":{"rendered":"Algebra Challenge &#8211; 3"},"content":{"rendered":"\n<p>For integer $n&gt;1$, find the value of $$\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$$ Click <a onclick=\"toggle_visibility('algebra-chall-1');\">here<\/a> for the solution.\n<\/p>\n\n\n\n<div id=\"algebra-chall-1\" style=\"display:none\">\n\n\n\n<p><strong>Solution:<\/strong> $$\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}-\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$$<\/p>\n\n\n\n<p>$=1+\\dfrac{\\sum_{i=1}^{n^2-1}(\\sqrt{n+\\sqrt{i}}-\\sqrt{n-\\sqrt{i}})}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{(\\sqrt{n+\\sqrt{i}}-\\sqrt{n-\\sqrt{i}})^2}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$<\/p>\n\n\n\n<p>$=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{2n-2\\sqrt{n^2-i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sqrt{2}\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{n^2-i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$<\/p>\n\n\n\n<p>$=1+\\dfrac{\\sqrt{2}\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=\\boxed{1+\\sqrt{2}}$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>For integer $n&gt;1$, find the value of $$\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$$ Click here for the solution. Solution: $$\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{n+\\sqrt{i}}-\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$$ $=1+\\dfrac{\\sum_{i=1}^{n^2-1}(\\sqrt{n+\\sqrt{i}}-\\sqrt{n-\\sqrt{i}})}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{(\\sqrt{n+\\sqrt{i}}-\\sqrt{n-\\sqrt{i}})^2}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$ $=1+\\dfrac{\\sum_{i=1}^{n^2-1}\\sqrt{2n-2\\sqrt{n^2-i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=1+\\dfrac{\\sqrt{2}\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{n^2-i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}$ $=1+\\dfrac{\\sqrt{2}\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}{\\sum_{i=1}^{n^2-1}\\sqrt{n-\\sqrt{i}}}=\\boxed{1+\\sqrt{2}}$<\/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":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4120"}],"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=4120"}],"version-history":[{"count":27,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4120\/revisions"}],"predecessor-version":[{"id":4888,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4120\/revisions\/4888"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}