{"id":3155,"date":"2012-08-23T23:07:34","date_gmt":"2012-08-23T23:07:34","guid":{"rendered":"http:\/\/schooltutoring.com\/scholarship\/?p=3155"},"modified":"2014-12-02T08:32:05","modified_gmt":"2014-12-02T08:32:05","slug":"arithmetic-and-geometric-means","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2012\/08\/23\/arithmetic-and-geometric-means\/","title":{"rendered":"Arithmetic and Geometric Means"},"content":{"rendered":"

For any two numbers a<\/em><\/strong> and b<\/em><\/strong>, we can insert a number A<\/em><\/strong> between them such that a,A, b<\/strong> are in arithmetic progression. \u00a0Here, A<\/strong> is called the arithmetic mean of a<\/strong> and b<\/strong>. Since a,A,b<\/strong> are in arithmetic progression, we have<\/p>\n

A-a = b-A<\/strong><\/p>\n

2A = a+b<\/strong><\/p>\n

A = (a+b)\/2.<\/strong><\/p>\n

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

Find the arithmetic mean of 16<\/strong> and 30<\/strong>.<\/p>\n

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

Here a=16<\/strong> and b=30.<\/strong><\/p>\n

Arithmetic mean = (a+b)\/2 = (16+30)\/2 = 46\/2 = 23.<\/strong><\/p>\n

Geometric mean:<\/h5>\n

For any two numbers a<\/em><\/strong> and b<\/em><\/strong>, we can insert a number G<\/em><\/strong> between them such that a, <\/em>G, b<\/strong> are in geometric progression.\u00a0 Here, G<\/em><\/strong> is called the arithmetic mean of a<\/strong> and b<\/strong>. Since a,G,b<\/strong> are in geometric progression, we have<\/p>\n

G\/a = b\/G<\/strong><\/p>\n

G2<\/sup> = ab<\/strong><\/p>\n

G = \u221aab<\/strong><\/p>\n

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

Find the geometric mean of 9<\/strong> and 16<\/strong>.<\/p>\n

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

Here a=9<\/strong> and b=16.<\/strong><\/p>\n

 <\/p>\n

Geometric mean = \u221aab=\u221a9*16=\u221a144 = 12.<\/strong><\/p>\n

Relation between arithmetic and geometric means:<\/h4>\n

Let A<\/strong> and G<\/strong> be the arithmetic and geometric means respectively of two numbers a<\/strong> and b<\/strong>.<\/p>\n

Then<\/p>\n

A = (a+b)\/2<\/strong><\/p>\n

G= \u221aab.<\/strong><\/p>\n

A-G = (a+b)\/2 – \u221aab<\/strong><\/p>\n

= (a+b-2\u221aab)\/2<\/strong><\/p>\n

= (\u221aa-\u221ab)2<\/sup>\/2,<\/strong><\/p>\n

which is always non negative. (\u22650)<\/strong>.<\/p>\n

So, A-G \u22650<\/strong><\/p>\n

i.e. A \u2265 G.<\/strong><\/p>\n

Still need help with Mathematics? Please read more about our Mathematics tutoring services<\/a>.<\/p>\n

SchoolTutoring Academy<\/span> 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":"

For any two numbers a and b, we can insert a number A between them such that a,A, b are in arithmetic progression. \u00a0Here, A is called the arithmetic mean of a and b. Since a,A,b are in arithmetic progression, we have A-a = b-A 2A = a+b A = (a+b)\/2. Example: Find the arithmetic […]<\/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":[793,854,2035,2161,2194],"class_list":["post-3155","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-how-do-aritmetic-and-geometric-means-interact","tag-how-does-aritmetic-progression-work","tag-what-are-means","tag-what-is-a-geometric-mean","tag-what-is-an-arithmetic-mean"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/3155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=3155"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/3155\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=3155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=3155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=3155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}