Problem 50953. Round up to π

Created by ChrisR

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{"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:t>To commemorate Pi Day (March 14 in the U.S.), compute the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>th term in a sequence by starting with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>, rounding up to the next multiple of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n-1\"/></w:customXmlPr><w:r><w:t>n-1</w:t></w:r></w:customXml><w:r><w:t>, rounding up to the next multiple of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n-2\"/></w:customXmlPr><w:r><w:t>n-2</w:t></w:r></w:customXml><w:r><w:t>, and so on to rounding up to the next multiple of 1. For example, with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> = 8, you would get 14, 18, 20, 20, 21, 22, and—the element in the sequence—</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"f(n)\"/></w:customXmlPr><w:r><w:t>f(n)</w:t></w:r></w:customXml><w:r><w:t> = 22. </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>Write a function that computes </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"f(n)\"/></w:customXmlPr><w:r><w:t>f(n)</w:t></w:r></w:customXml><w:r><w:t>, the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>th term in this sequence. Also compute </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n^2/f(n)\"/></w:customXmlPr><w:r><w:t>n^2/f(n)</w:t></w:r></w:customXml><w:r><w:t> and notice its limit as </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> gets large. For more on this limit, see </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathpages.com/home/kmath001/kmath001.htm\"><w:r><w:rPr><w:u/></w:rPr><w:t>this page</w:t></w:r></w:hyperlink><w:r><w:t> by K.S. Brown.</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: 20.4333px; 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: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; "><div style="block-size: 63px; 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 31.5px; text-align: left; transform-origin: 384px 31.5px; 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: 191.35px 7.91667px; transform-origin: 191.35px 7.91667px; unicode-bidi: normal; ">To commemorate Pi Day (March 14 in the U.S.), compute the n<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: 117.85px 7.91667px; transform-origin: 117.85px 7.91667px; unicode-bidi: normal; ">th term in a sequence by starting with n<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: 62.2333px 7.91667px; transform-origin: 62.2333px 7.91667px; unicode-bidi: normal; ">, rounding up to the next multiple of <img src="data:image/png;base64,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" alt="n-1" style="width: 35px; height: 18px;" width="35" height="18"><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: 111.242px 7.91667px; transform-origin: 111.242px 7.91667px; unicode-bidi: normal; ">, rounding up to the next multiple of <img src="data:image/png;base64,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" alt="n-2" style="width: 35px; height: 18px;" width="35" height="18"><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: 171.908px 7.91667px; transform-origin: 171.908px 7.91667px; unicode-bidi: normal; ">, and so on to rounding up to the next multiple of 1. For example, with n<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: 244.467px 7.91667px; transform-origin: 244.467px 7.91667px; unicode-bidi: normal; "> = 8, you would get 14, 18, 20, 20, 21, 22, and—the element in the sequence—<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAmCAYAAACbBvanAAACiklEQVRoge2Zb5GDMBDFfx5wUAM1gIIqqAMc1EEtVEMlnAcsoKEWuA/JG/bugGxC4MvxZjod2slms/v2X4ATJ06cOLEbLkCbuaYBXvG7lg7PivLcuAID8EU40AfoHOuauOZaWZ8W6DnQEC0wAu/43MXnT0KJhqBoLnO8uEf5u6MhHPZDoCHAjWCEr8TaN4G2e+J1wB48+MkCL+6kmVIDctJebIO4wUg4VA4GggGPwDPutwvuBAOMTKGQs+6opHWJ+932EP6OwnOt3JPOF7XRkx+yi+gINH4whcJgfutY97C8khMKV4IXrScbAqMe+Dz8jPtWgQ77YgqFt/k9lRsUCqlE1RHYMpqPZN+YHKBPyhDefbOgqpArWOu8zVHPz57jTmBey1SKR9JlUL2Mp4FzQ/kgt8zJu14MTGxr47P208E8jriSH4ZJiI65CS7HCMof8qAYIIhVXkeU9DMu5XK7sRwj2BI88JfKYqP3YCVOW4SNxdzam2MEHXJJecsSD6oaQeWmpOHJMYJC7sPfRGrzgadRKynNq9BBSiY0GTBlPHvI14ocb6MmedWMsKZcChqzU9ncsm2unKp0Kiel2KAQrtInqNSUDE12fSqOdcg5T9vE3OK7mFElqTKvyJO5Q5PFwDqLGrPHHH1/69CTdkhPxQsWtctbRtMnIdktwZbGOfra/z2sklFLmDuL37FYAtF5SamWaRZZggY5Dxs7Kl7gWJpuvRh9ctw4PddoFUM0rBFbDUG5ahRdwIMN+updgLWgOrha46iu6UsTrEe+vQDOhh2VG6bSVtIbrOFGCIvaV226zt8UtrKiKJsqa1v3qmmIhsDaKi90LoRw6NiPskJNJjSV5Z04ceLE/8Q372/pj3B4BJgAAAAASUVORK5CYII=" alt="f(n)" style="width: 32.5px; height: 19px;" width="32.5" height="19"><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: 19.6417px 7.91667px; transform-origin: 19.6417px 7.91667px; unicode-bidi: normal; "> = 22. </div><div style="block-size: 42px; 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 21px; text-align: left; transform-origin: 384px 21px; 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: 96.325px 7.91667px; transform-origin: 96.325px 7.91667px; unicode-bidi: normal; ">Write a function that computes <img src="data:image/png;base64,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" alt="f(n)" style="width: 32.5px; height: 19px;" width="32.5" height="19"><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: 15.55px 7.91667px; transform-origin: 15.55px 7.91667px; unicode-bidi: normal; ">, the n<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: 122.517px 7.91667px; transform-origin: 122.517px 7.91667px; unicode-bidi: normal; ">th term in this sequence. Also compute <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAoCAYAAAD9htTzAAADhklEQVR4nO2a7bGqMBCG3x7owAZowAqswA7swA5ogRoowR5swRpowfMjvJeVGw7ZD9R7J88Mc+aohGR3s18BqFQqlUrl/+YI4Dpdxw/PpbKgAXAH8Jj+PqdrmL6rfAE3ABfxfwtgRFLU1Thm651UZeaAtHuWXJCUlPtuiwZJyZ90mS3ivECDJKePcUDe6o+wK+mM5Dq9nAH0mHd6qdB5XxQNkus/B44ZApXUGe4djPcRCoW78YRyg4lWkOSOL1PUFUkw2thyMN4n6acxTou5jPh9N52QdvBeyU4zjf818fYB2264wOfqpJulsBskRZ3WbkIyjnfEwb0NoRiPoO94zRS13GDLKhm73sENPnfupsWrFWugq7NmQrz/Cd2O4H3vyibP8K3TBYta62I7+Ky5Q1nsWdIjJpvUMMJeQ5ppkAScU1BpRvNQ/DYHux3atH/EfhndGgMMhnHA3HsbkBYqNX2dBl0GZWDeQblYckGZxTDga90kOxtr15br5XO3jMMjnxzMOFUu7zTdKBfNqvuOtEsG8Z2coPw8d5VMpIfPmunnZepdAoW15aI98slB49DM9Q/Mju6YWz7cIRz4KT5rMXe+c1ep+xqtE55gPNJaJ2uq0h2slc8a/K0pLklL6PGaKkprjcxMzkhK8iDjkcZlUuilRMmngbEjIy2BdYNcMK3VmmKv4W0DAa/z1qBRUrR8njBks9JljPjbGhgYIwsxWpSnVaJxM0s0SoqWj0lJ8sBuuVhrobiFtw3EMWQw16BRUrR81DtfPmS5jYHZ32oLxS28bSBgzqosPTHeu0W0fEyJgwx6OaFxMYNm0A28bSBCN2PpVpR266PlU1qfZR+yZgmsDzjBC/zC7WA7FJQwplnTWZ41baX/0fKhi1a5Zz4kZwlcCK3+jJiXTLxtIOB1bpZYSSVvKThaPj2UZUcrHpIT2lV832Mu5DzwmV5Flx7o/QbbPGvsIR91v1A+JDc4D8Q4iYhCtkdMfGN25umebx0dRMuH8ehrTmjX8LaB5DjqAJzBepJs4Z0HjGZO8LeBgNkNRZQFR+QL1Gg454++4lXCAFvHe7kwdgCidkCH2BIjR0RduDvWNhAzKLkDH4h/qWPAfkLs8P7DRRPWFx9ZV1ApHfZ5u6dBEmT0+3F7vs8XzvJEUwMVc0da8J7Z0RVxO/SAf8DFkag2UGVHWnz4fbNKpVKpfB0/VsODsOZTZz8AAAAASUVORK5CYII=" alt="n^2/f(n)" style="width: 52.5px; height: 20px;" width="52.5" height="20"><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: 68.8417px 7.91667px; transform-origin: 68.8417px 7.91667px; unicode-bidi: normal; "> and notice its limit as n<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: 17.1083px 7.91667px; transform-origin: 17.1083px 7.91667px; unicode-bidi: normal; "> gets large. For more on this limit, see <a target='_blank' href = "https://www.mathpages.com/home/kmath001/kmath001.htm">this page</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: 48.2167px 7.91667px; transform-origin: 48.2167px 7.91667px; unicode-bidi: normal; "> by K.S. Brown.</div></div></div>

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Problem 50953. Round up to π

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