{"id":1290,"date":"2016-10-16T06:21:54","date_gmt":"2016-10-16T06:21:54","guid":{"rendered":"http:\/\/www.testingdocs.com\/questions\/?p=1290"},"modified":"2024-12-24T16:26:26","modified_gmt":"2024-12-24T16:26:26","slug":"what-is-mathematical-induction","status":"publish","type":"post","link":"https:\/\/www.testingdocs.com\/questions\/what-is-mathematical-induction\/","title":{"rendered":"What is Mathematical Induction?"},"content":{"rendered":"<h2>What is Mathematical Induction?<\/h2>\n<p><strong>Mathematical induction<\/strong> is a method to prove that some statement about the integer n is true for all n. In this method, first, we prove the statement when n has its smallest value, say p0. This is called the base case or basis or trivial case.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2116 size-full\" title=\"What is Mathematical Induction?\" src=\"http:\/\/www.testingdocs.com\/questions\/wp-content\/uploads\/Mathematical-Induction.png\" alt=\"What is Mathematical Induction?\" width=\"1104\" height=\"612\" \/><\/p>\n<p>Then you prove the statement for k &gt; p0, assuming that it has already been proved for all values between p0 and k, i.e You will assume that<strong> P(k)<\/strong> holds true. With that assumption we will prove the case for <strong>P(k+1)<\/strong> . This is called the induction.<\/p>\n<p>Please see the above picture diagram for general idea about <b>mathematical<\/b> induction. This kind of proof is ideal for recurrences.<\/p>\n<h3>Principle of <b>Mathematical<\/b> Induction<\/h3>\n<p>The principle of <b>mathematical<\/b> induction is usually described in two forms. The one we have talked about is called the weak form.<\/p>\n<p>If the statement p(base) holds true, and the statement P(k) \u21d2 P(k+1) holds true for all k &gt; base, then<\/p>\n<p>P(n) is true for all integers n \u2265 base.<\/p>\n<p>Mathematical induction is a method of proving that a statement is true for all natural numbers. It\u2019s particularly useful when dealing with sequences or series, and it relies on a two-step process:<\/p>\n<p><strong>Base Case<\/strong>: Verify that the statement is true for the initial value, usually<strong> n=1<\/strong>. This establishes the starting point of the induction.<\/p>\n<p><strong>Inductive Step<\/strong>: Assume that the statement is true for some arbitrary natural number<br \/>\n<strong>k<\/strong>. This assumption is known as the &#8220;inductive hypothesis.&#8221; Then, you need to show that if the statement holds for <strong>k<\/strong>, it must also hold for <strong>k+1<\/strong>.<\/p>\n<h4><strong>Strong form:<\/strong><\/h4>\n<p>If the statement p(base) holds true, and the statement<\/p>\n<p>P(base)\u2227 P(b+1) \u2227 \u2026 P(k)\u21d2 P(k+1)) holds true for all k &gt; base, then<\/p>\n<p>P(n) is true for all integers n \u2265 base.<\/p>\n<h2>Example<\/h2>\n<p>Some of the examples are as follows:<\/p>\n<p>#1<\/p>\n<p>Using mathematical induction, let&#8217;s prove the formula for the Sum of N natural number squares.<\/p>\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Mathematical Induction\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\LARGE&amp;space;1^2&amp;space;+&amp;space;2^2&amp;space;+&amp;space;3^2&amp;space;+&amp;space;...&amp;space;+&amp;space;n^2&amp;space;=&amp;space;\\frac{n(n+1)(2n+1)}{6}\" alt=\"Mathematical Induction\" width=\"539\" height=\"67\" align=\"absmiddle\" \/><\/h3>\n<ul>\n<li><a href=\"https:\/\/www.testingdocs.com\/questions\/sum-of-n-natural-number-squares\/\">https:\/\/www.testingdocs.com\/questions\/sum-of-n-natural-number-squares\/<\/a><\/li>\n<\/ul>\n<p>#2<\/p>\n<p>Another example of formula proof using mathematical induction:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\LARGE&amp;space;\\prod_{i=2}^{n}\\left&amp;space;(&amp;space;1&amp;space;-&amp;space;\\frac{1}{i^{2}}&amp;space;\\right&amp;space;)=&amp;space;\\frac{n+1}{2n}\" alt=\"\\LARGE \\prod_{i=2}^{n}\\left ( 1 - \\frac{1}{i^{2}} \\right )= \\frac{n+1}{2n}\" align=\"absmiddle\" title=\"\"><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li><a href=\"https:\/\/www.testingdocs.com\/questions\/use-mathematical-induction-to-prove-the-statement\/\">https:\/\/www.testingdocs.com\/questions\/use-mathematical-induction-to-prove-the-statement\/<\/a><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is Mathematical Induction? Mathematical induction is a method to prove that some statement about the integer n is true for all n. In this method, first, we prove the statement when n has its smallest value, say p0. This is called the base case or basis or trivial case. &nbsp; Then you prove the [&hellip;]<\/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-1290","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\/1290","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=1290"}],"version-history":[{"count":16,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/1290\/revisions"}],"predecessor-version":[{"id":26514,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/1290\/revisions\/26514"}],"wp:attachment":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/media?parent=1290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/categories?post=1290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/tags?post=1290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}