Problem 58274. Easy Sequences 116: Summation involving nthroot, floor, and ceiling functions

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a postive integers </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, we are asked to evaluate the following summation:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>s= S(x,n) = \\sum_{i=1}^x \\bigg(\\Big\\lceil \\sqrt[n]{i -1} \\Big\\rceil+\\Big\\lfloor \\sqrt[n]{i+1} \\Big\\rfloor\\bigg)</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                            where: the symbol \"</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\lfloor\\ \\rfloor</w:t></w:r></w:customXml><w:r><w:t>\" and \"</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\lceil\\ \\rceil</w:t></w:r></w:customXml><w:r><w:t>\" are the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/floor.html#\"><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>floor</w:t></w:r></w:hyperlink><w:r><w:t> and </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/ceil.html\"><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>ceil</w:t></w:r></w:hyperlink><w:r><w:t> functions, respectively.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example, if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=50</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=3</w:t></w:r></w:customXml><w:r><w:t>, we have:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[>>  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\n>>  s = S(50,3)\n>>  s =\n        279]]></w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 226px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 113px; transform-origin: 407px 113px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Given a postive integers </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; "> and </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, we are asked to evaluate the following summation:</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                            </span></span><span style="vertical-align:-17px"><img 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" width="266.5" height="46" style="width: 266.5px; height: 46px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                            where: the symbol "</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="23.5" height="19" style="width: 23.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">" and "</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="23.5" height="19" style="width: 23.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">" are the </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/ref/floor.html#"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">floor</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/ref/ceil.html"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">ceil</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> functions, respectively.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">For example, if </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="44" height="18" style="width: 44px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we have:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40px; transform-origin: 404px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  s = S(50,3)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  s =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">        279</span></span></div></div></div></div>

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Problem 58274. Easy Sequences 116: Summation involving nthroot, floor, and ceiling functions

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Problem 58274. Easy Sequences 116: Summation involving nthroot, floor, and ceiling functions

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