{"id":591,"date":"2020-03-17T20:35:24","date_gmt":"2020-03-17T20:35:24","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=591"},"modified":"2024-10-25T02:38:32","modified_gmt":"2024-10-25T06:38:32","slug":"telescopic-method-for-summary","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=591","title":{"rendered":"Telescopic Method for Summary and Product"},"content":{"rendered":"\n

The Telescopic Method<\/strong> is a technique for calculating the summary or product of a certain series in which each term can be decomposed into multiple parts, with some of them cancelling those of the next term. <\/p>\n\n\n\n

For example, to calculate the following summary, in which each term is decomposed into two parts:<\/p>\n\n\n\n

$$\\begin{align}\nS&=\\dfrac{1}{1\\times 2}+\\dfrac{1}{2\\times 3}+\\dfrac{1}{3\\times 4}+…+\\dfrac{1}{99\\times 100} \\\\\n&=\\left(\\dfrac{1}{1}-\\cancel{\\dfrac{1}{2}}\\right)+\\left(\\cancel{\\dfrac{1}{2}}-\\cancel{\\dfrac{1}{3}}\\right)+\\left(\\cancel{\\dfrac{1}{3}}-\\cancel{\\dfrac{1}{4}}\\right)+…+\\left(\\cancel{\\dfrac{1}{99}}-\\dfrac{1}{100}\\right) \\\\\n&=\\dfrac{1}{1}-\\dfrac{1}{100}=\\dfrac{99}{100} \\\\\n\\end{align}\n$$\n<\/p>\n\n\n\n

The similar technique can be used for calculating the product of a certain series, such as:<\/p>\n\n\n\n

$$\\begin{align}\nP&=\\left(1-\\dfrac{2}{3}\\right)\\times\\left(1-\\dfrac{2}{5}\\right)\\times\\left(1-\\dfrac{2}{7}\\right)\\times … \\times\\left(1-\\dfrac{2}{97}\\right)\\times\\left(1-\\dfrac{2}{99}\\right) \\\\\n&=\\dfrac{1}{\\cancel{3}}\\times\\dfrac{\\cancel{3}}{\\cancel{5}}\\times\\dfrac{\\cancel{5}}{\\cancel{7}}\\times … \\times\\dfrac{\\cancel{95}}{\\cancel{97}}\\times\\dfrac{\\cancel{97}}{99}=\\dfrac{1}{99} \\\\\n\\end{align}\n$$\n<\/p>\n\n\n\n

Find the answers for the following questions:<\/p>\n\n\n\n


  1. $S=\\dfrac{1}{3^2-1}+\\dfrac{1}{5^2-1}+\\dfrac{1}{7^2-1}+…+\\dfrac{1}{99^2-1}$<\/li>

  2. $S=\\dfrac{1}{1\\times 2\\times 3}+\\dfrac{1}{2\\times 3\\times 4}+\\dfrac{1}{3\\times 4\\times 5}+…+\\dfrac{1}{98\\times 99\\times 100}$<\/li>

  3. $S=\\dfrac{1}{\\sqrt{1}+\\sqrt{2}}+\\dfrac{1}{\\sqrt{2}+\\sqrt{3}}+\\dfrac{1}{\\sqrt{3}+\\sqrt{4}}+…+\\dfrac{1}{\\sqrt{99}+\\sqrt{100}}$<\/li>

  4. $S=\\dfrac{2^2+1}{2^2-1}+\\dfrac{4^2+1}{4^2-1}+\\dfrac{6^2+1}{6^2-1}+…+\\dfrac{100^2+1}{100^2-1}$<\/li>

  5. $S=\\left(1-\\dfrac{1}{2}\\right)\\times\\left(1-\\dfrac{1}{3}\\right)\\times\\left(1-\\dfrac{1}{4}\\right)\\times … \\times\\left(1-\\dfrac{1}{100}\\right)$<\/li>

  6. $P=\\left(1-\\dfrac{1}{2^2}\\right)\\times\\left(1-\\dfrac{1}{3^2}\\right)\\times\\left(1-\\dfrac{1}{4^2}\\right)\\times … \\times\\left(1-\\dfrac{1}{100^2}\\right)$<\/li>

  7. $P=\\dfrac{2^3+1}{2^3-1}\\times\\dfrac{3^3+1}{3^3-1}\\times\\dfrac{4^3+1}{4^3-1}\\times…\\times\\dfrac{100^3+1}{100^3-1}$<\/li>

  8. $S=\\sum_{n=1}^{100}\\left(\\dfrac{4n}{4n^4+1}\\right)$<\/li>

  9. $S=\\sum_{n=1}^{100}\\left(\\dfrac{n^2-\\dfrac{1}{2}}{n^4+\\dfrac{1}{4}}\\right)$<\/li>

  10. $S=1\\times 1!+2\\times 2!+3\\times 3!+4\\times 4!+…100\\times 100!$<\/li>

  11. $S=\\dfrac{1}{2\\sqrt{1}+1\\sqrt{2}}+\\dfrac{1}{3\\sqrt{2}+2\\sqrt{3}}+\\dfrac{1}{4\\sqrt{3}+3\\sqrt{4}}+…+\\dfrac{1}{24\\sqrt{23}+23\\sqrt{24}}+\\dfrac{1}{25\\sqrt{24}+24\\sqrt{25}}$<\/li>

  12. $P=\\dfrac{3^2-1}{3^2-4}\\times\\dfrac{4^2-1}{4^2-4}\\times\\dfrac{5^2-1}{5^2-4}\\times … \\times\\dfrac{100^2-1}{100^2-4}$<\/li>

  13. $P=\\dfrac{2^2}{2^2-1}\\times\\dfrac{3^2}{3^2-1}\\times\\dfrac{4^2}{4^2-1}\\times … \\times\\dfrac{50^2}{50^2-1}$<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

    The Telescopic Method is a technique for calculating the summary or product of a certain series in which each term can be decomposed into multiple parts, with some of them cancelling those of the next term. For example, to calculate … Continue reading →<\/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":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/591"}],"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=591"}],"version-history":[{"count":46,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/591\/revisions"}],"predecessor-version":[{"id":5575,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/591\/revisions\/5575"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}