{"id":1357,"date":"2016-07-21T10:16:22","date_gmt":"2016-07-21T10:16:22","guid":{"rendered":"http:\/\/www.testingdocs.com\/questions\/?p=1357"},"modified":"2024-12-13T15:11:45","modified_gmt":"2024-12-13T15:11:45","slug":"sum-of-n-natural-number-squares","status":"publish","type":"post","link":"https:\/\/www.testingdocs.com\/questions\/sum-of-n-natural-number-squares\/","title":{"rendered":"Mathematical Induction : Sum of N natural number squares"},"content":{"rendered":"

Mathematical Induction : Sum of N natural number squares<\/h2>\n

Using Mathematical Induction<\/a> Prove that the sum of the first n natural number squares is:<\/p>\n

 <\/p>\n

\"\\LARGE<\/p>\n

i.e<\/p>\n

<\/h3>\n

\"\\LARGE<\/h3>\n

Using WolframAlpha to find the formula. WolframAlpha is a great mathematical computational tool for Mathematical students.<\/p>\n

    \n
  • https:\/\/www.wolframalpha.com\/<\/strong><\/li>\n<\/ul>\n

     <\/p>\n

    \"WolframAlpha_Sum_of_Squares\"<\/p>\n

     <\/p>\n

     <\/p>\n

    Proof<\/h2>\n

    By Mathematical Induction.<\/p>\n

    Base case : P(1)<\/h3>\n

    Let’s prove that the base case holds true. We need to show that P(1) is true.<\/p>\n

    \"\\LARGE<\/p>\n

    \"\\LARGE<\/p>\n

    1=1<\/p>\n

    P(1) is true.<\/p>\n

    Induction Case<\/h3>\n

    Let’s assume that P(k) hold true. Assume that for some positive k, P(k) holds true, so that<\/p>\n

    P(k)<\/p>\n

    \"\\LARGE<\/p>\n

    We need to show that P(k + 1) holds true.<\/p>\n

    \"\\LARGE<\/p>\n

    We have assumed that P(k) is true. Replacing\u00a0 for k terms we get:<\/p>\n

    \"\\LARGE<\/p>\n

    Taking (k+1) common we get :<\/p>\n

    \"\\LARGE<\/p>\n

    Take hints in the factorization, we need to prove and eventually get (k+1) in terms of k.<\/p>\n

    expand and rearrange the 2nd term:<\/p>\n

    \"\\LARGE<\/p>\n

    \"\\LARGE<\/p>\n

    \"\\LARGE<\/p>\n

    Taking k+2 common<\/p>\n

    \"\\LARGE<\/p>\n

    \"\\LARGE<\/p>\n

     <\/p>\n

    Hence P(k + 1) holds true, when P(k) is true, so P(n) is true for all finite natural numbers n.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Mathematical Induction : Sum of N natural number squares Using Mathematical Induction Prove that the sum of the first n natural number squares is:   i.e Using WolframAlpha to find the formula. WolframAlpha is a great mathematical computational tool for Mathematical students. https:\/\/www.wolframalpha.com\/       Proof By Mathematical Induction. Base case : P(1) Let’s […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[849],"tags":[],"class_list":["post-1357","post","type-post","status-publish","format-standard","hentry","category-math-questions","has-post-title","has-post-date","has-post-category","has-post-tag","has-post-comment","has-post-author",""],"_links":{"self":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/1357","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/comments?post=1357"}],"version-history":[{"count":17,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/1357\/revisions"}],"predecessor-version":[{"id":26308,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/1357\/revisions\/26308"}],"wp:attachment":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/media?parent=1357"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/categories?post=1357"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/tags?post=1357"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}