{"id":7911,"date":"2022-05-11T14:28:16","date_gmt":"2022-05-11T08:58:16","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7911"},"modified":"2025-06-02T20:46:02","modified_gmt":"2025-06-02T15:16:02","slug":"what-is-the-principle-of-mathematical-induction","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-principle-of-mathematical-induction\/","title":{"rendered":"What is the principle of mathematical induction?"},"content":{"rendered":"\n<p>Mathematical induction is a mathematical technique is used to prove a statement, a theorem or a formula is true for every natural number.<\/p>\n\n\n\n<p>An essential property of the set <strong>N <\/strong>= {1, 2, 3, \u2026} of positive integers follows:<\/p>\n\n\n\n<p><strong>Principle of Mathematical Induction I: <\/strong>Let P be a proposition defined on the positive integers <strong>N<\/strong>; that is, P(n) is either true or false for each n \u2208 <strong>N<\/strong>. Suppose P has the following two properties:<\/p>\n\n\n\n<p>(i) P(1) is true.<\/p>\n\n\n\n<p>(ii) P(k + 1) is true whenever P(k) is true.<\/p>\n\n\n\n<p>Then P is true for every positive integer n \u2208 <strong>N<\/strong>.<\/p>\n\n\n\n<p>We shall not prove this principle. In fact, this principle is usually given as one of the axioms when <strong>N <\/strong>is developed axiomatically.<\/p>\n\n\n\n<p><strong>Principle of Mathematical Induction II: <\/strong>Let P be a proposition defined on the positive integers <strong>N <\/strong>such that:<\/p>\n\n\n\n<p>(i) P(1) is true.<\/p>\n\n\n\n<p>(ii) P(k) is true whenever P(j) is true for all 1 \u2264 j &lt; k.<\/p>\n\n\n\n<p>Then P is true for every positive integer n \u2208 <strong>N<\/strong>.<\/p>\n\n\n\n<p><strong>Note: <\/strong>Sometimes one wants to prove that a proposition P is true for the set of integers {a, a + 1, a + 2, a + 3, . . .} where a is any integer, possibly zero. This can be done by simply replacing 1 by a in either of the above Principles of Mathematical Induction.<\/p>\n\n\n\n<p>The technique has two steps to prove any statement-<\/p>\n\n\n\n<p>1. Base step (it proves that a statement is true for the initial value)<\/p>\n\n\n\n<p>2. Inductive step (it proves that is the statement is true for n\u2019th iteration then it is also true for&nbsp; (n+1)\u2019th iteration.<\/p>\n\n\n\n<p>Example: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/kq3SOWv_OsM_SeWaQb6CRMousAyR_KkqPis4lUlPs9VuUBGsoaFgpRrZyelI48_FLT5xHo9MpwUFGB75B5oxlft9oqglCjqAWj46Dm8jWWtcbHCal8zI_-nvN-C5dyNxt6rEviPFi4NgLvnQBQ\" width=\"441\" height=\"24\"><\/p>\n\n\n\n<p>Sol. here for n = 1, 1 = 1\u00b2<\/p>\n\n\n\n<p>Now let us assume that the statement is true for n = k&nbsp;<\/p>\n\n\n\n<p>Hence we assume that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/d2dRhz2RUraqkSvIR9mL0_j_PvFgLyRnAfPaKkXIhBxYxr1pc0Q6Uh6ZT4XX1g9erIbnYQiqFi6wCU6D4EUghOBsG_E9fLPe7oQDQQjgaQ140WBblZCxM2U23DTjWM3r_S5BgG6hf6YaJxvuDw\" width=\"273\" height=\"31\"> is true.<\/p>\n\n\n\n<p>Now we need to prove that-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/JzEDj8tkio4I6-UZy3jnQRmjEk1NIOUvXBL54sFiJFdZPE4AX7egXAp-mS7Q9g2oyTHpkBY-FmWLaWYt5_Qe0QowwQAvcUnp77yI4ro342OUtN-y0AG4ezMxvBjpibNeVQC48KfK6sGgQg_jxA\" alt=\"\" \/><\/figure>\n\n\n\n<p>So that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/dCNES0gSF44nrQ3mXza-aphB2vBjkYH51eWdVfGUwF40QWMsvRnohcQ-ABDbCkdaW_m3ExqV1zEeohODEaqXkXXad0tr8BHYAhXFkfux_3dqYm1a9bObn9xYuB8AdrwmZLVnliXiCc9uSm1C6A\" width=\"369\" height=\"31\"> which satisfies the second step.<\/p>\n\n\n\n<p>Hence-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/1X2nfWkVd6Wuol0aBZku-fOhZSUqXfPUHY53dgwlH8awn3itfIhou-zDHvX5FtlosPHepKJ1AawWbvo5Y-n2ltsmGHcGm2SI2on0TupfKielCXyv8g9bz4bNIy-jguX31HMAVOwZptgHK0fMZA\" alt=\"\" \/><\/figure>\n\n\n\n<p><strong>Example-2:<\/strong><strong>Prove 1+2+&#8230;+n=n(n+1)\/2 using a proof by induction<\/strong><\/p>\n\n\n\n<p><strong>Sol.<\/strong><\/p>\n\n\n\n<p>Let n=1: 1 = 1(1+1)\/2 = 1(2)\/2 = 1 is true,<\/p>\n\n\n\n<p><strong>Step-1: <\/strong>Assume n=k holds: 1+2+&#8230;+k=k(k+1)\/2<\/p>\n\n\n\n<p>Show n=k+1 holds: 1+2+&#8230;+k+(k+1)=(k+1)((k+1)+1)\/2&nbsp;<\/p>\n\n\n\n<p>Substitute k with k+1 in the formula to get these lines. Notice that I write out what I want to prove.<\/p>\n\n\n\n<p><strong>Step-2: <\/strong>Now start with the left side of the equation<\/p>\n\n\n\n<p>1+2+&#8230;+(k+1)=1+2+&#8230;+k+(k+1)<\/p>\n\n\n\n<p>=k(k+1)\/2 + (k+1)<\/p>\n\n\n\n<p>=(k(k+1) + 2(k+1))\/2 by 2\/2=1 and distribution of division over addition<\/p>\n\n\n\n<p>=(k+2)(k+1)\/2 by distribution of multiplication over addition<\/p>\n\n\n\n<p>=(k+1)(k+2)\/2 by commutativity of multiplication.<\/p>\n\n\n\n<p><strong>Example-3: Prove the following by using the principle of mathematical induction for all n <\/strong><strong>\u2208<\/strong><strong> N-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/qZtmQmfxPfG8FIHRjvkVjhZvNmhr8ZyE4Sv5jEHhkE5eZ2bxdyPvGBqFkPAkVZ9w9kvOonPoT_TVEs5KoeoEv7zlz3-66K6vChAa_Gyd7_K2iAokmf_T7hs_hEG620AlKz2-gbZtybK9gXshCg\" alt=\"\" \/><\/figure>\n\n\n\n<p>Sol. Here, n = 1, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/D8f3waQYhwiFrfwvoYrP8HMbQtQ2Bk1EMOccPm0KWzFQvsQeztKThuCPmGa9awgt_GQCjv5Q3i-hWVzMo_2Ua5OAAQmknQzwjZZLFTttRAn1pcoJ5k0K0mT1GMqQwMjBFRTAB5eURC9oNcS4aQ\" width=\"195\" height=\"30\"> which is true<\/p>\n\n\n\n<p>Step-1: Assume n = k holds-&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/QXwHU9GxL06SNEBLi8zKuaZvb2I2MXp2n85LTkrWCr9VwrGparENhCsf1WlcbIhJeoYQC4grQLBmfVATBAuYfcN8hn-TPpFn8szjKUaIhvupMO0hD16Jm7MIMAkrqmwuv5GS5Xzt0XfumOS0EQ\" alt=\"\" \/><\/figure>\n\n\n\n<p>Now show n = k + 1 also holds-<\/p>\n\n\n\n<p>Consider-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/jXfd6Wvoc2WrEB7KoTyhp0LWGJNn4Z0oBSXUc_75kq8C1xIuCznsTdUzvs2DG8vYUHJk3osP4T7QKMniYezlx2eFXzGio2Xr8uOVEPX08egU6GvzKF7EIH4gtsw96gf38uP3ZMSSuyfe2AdiQg\" alt=\"\" \/><\/figure>\n\n\n\n<p>Which is also true for n = k + 1.<\/p>\n\n\n\n<p>Hence proved.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mathematical induction is a mathematical technique is used to prove a statement, a theorem or a formula is true for every natural&hellip;<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-7911","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is the principle of mathematical induction? 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