{"id":3076,"date":"2012-08-13T22:30:10","date_gmt":"2012-08-13T22:30:10","guid":{"rendered":"http:\/\/SchoolTutoring.com\/help\/?p=3076"},"modified":"2014-12-02T08:32:06","modified_gmt":"2014-12-02T08:32:06","slug":"solving-theorems-mathematical-induction","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/solving-theorems-mathematical-induction\/","title":{"rendered":"Solving Theorems: Mathematical Induction"},"content":{"rendered":"

In Mathematics, we see many theorems which are true for all natural numbers. Sometimes, proving these theorems directly is not possible. For this purpose, we use a method of proof<\/a> which is known as \u201cMathematical Induction\u201d.<\/p>\n

The principal of Mathematical Induction is defined as follows.<\/p>\n

\u201cLet S(n) <\/em><\/strong>be a certain statement involving n<\/em><\/strong>, where n<\/em><\/strong> is a natural number.<\/p>\n

If S(1) <\/em><\/strong>is true and if S(k) <\/em><\/strong>implies S(k+1) <\/em><\/strong>where k <\/em><\/strong>is a natural number, then S(n) <\/em><\/strong>is true for all natural numbers n<\/em><\/strong>.\u201d<\/p>\n

Explanation:<\/h5>\n

The principle of mathematical induction can be explained in simple words as follows.<\/p>\n

Step 1: <\/strong><\/span>Prove S(1).<\/em><\/p>\n

Step 2:<\/strong> Assume that S(k)<\/em> is true doe some natural number k<\/em>.<\/p>\n

Step 3:<\/span> <\/strong>\u00a0Prove S(k+1).<\/em><\/p>\n

From step-1, S(n) is true for n=1. From step-3, S(n) is true for n=k+1, then automatically S(n) is true for n=1+1=2.<\/p>\n

Also, S(n) is true for n=2+1=3.<\/p>\n

Similarly S(n) is true for n=4,5,\u2026..<\/p>\n

So, by this principle of Mathematical induction, we can say that S(n) is true for all natural numbers.<\/p>\n

Example 1:<\/span><\/strong><\/p>\n

By the principle of mathematical induction, prove that<\/p>\n

1+2+3+\u2026+n = n(n+1)\/2.<\/p>\n

Solution:<\/strong><\/em><\/p>\n

Let S(n) : 1+2+3+\u2026+n = n(n+1)\/2.<\/p>\n

Step 1: <\/strong><\/em>Prove S(1).<\/p>\n

When n=1,<\/p>\n

LHS = 1, RHS = 1(1+1)\/2=1.<\/p>\n

LHS=RHS<\/p>\n

So, S(1) is proved.<\/p>\n

Step 2:<\/strong><\/em> Assume that S(k) is true doe some natural number k.<\/p>\n

1+2+3+\u2026+k = k(k+1)\/2.<\/p>\n

Step 3: <\/strong><\/em>\u00a0Prove S(k+1).<\/p>\n

LHS=1+2+3+\u2026+(k+1)<\/p>\n

=(1+2+3+\u2026+k) + (k+1)<\/p>\n

= k(k+1)\/2 + (k+1)<\/p>\n

= [k(k+1) + 2(k+1)]\/2<\/p>\n

=(k+1)(k+2)\/2<\/p>\n

=(k+1)[(k+1)+1]\/2<\/p>\n

=RHS.<\/p>\n

So, S(k+1) is true.<\/p>\n

So, by this principle of Mathematical induction, we can say that S(n) is true for all natural numbers.<\/p>\n

 <\/p>\n

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Palmdale visit: Tutoring in Palmdale . <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In Mathematics, we see many theorems which are true for all natural numbers. Sometimes, proving these theorems directly is not possible. For this purpose, we use a method of proof which is known as \u201cMathematical Induction\u201d. The principal of Mathematical Induction is defined as follows. \u201cLet S(n) be a certain statement involving n, where n […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[889,934,1182,1690,2341],"class_list":["post-3076","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-how-to-use-induction","tag-induction-in-math","tag-natural-number-theorems","tag-solving-math-indirectly","tag-what-is-induction"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3076","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=3076"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3076\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=3076"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=3076"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=3076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}