{"id":3247,"date":"2021-10-13T08:13:46","date_gmt":"2021-10-13T08:13:46","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=3247"},"modified":"2022-01-02T06:23:41","modified_gmt":"2022-01-02T06:23:41","slug":"sum-of-infinite-geometric-progression","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/sum-of-infinite-geometric-progression\/","title":{"rendered":"Sum of infinite geometric progression"},"content":{"rendered":"

[et_pb_section fb_built=”1″ module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.6″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|0px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.11.2″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n

Sum of infinite geometric progression<\/strong><\/h1>\n

[\/et_pb_text][et_pb_text _builder_version=”4.11.2″ _module_preset=”default” text_font_size=”16px” header_2_text_color=”gcid-70a40ab1-38f4-4842-b1ac-1e69b0045607″ header_3_text_color=”#777777″ custom_padding=”|15px||4px|false|false” global_colors_info=”{%22gcid-70a40ab1-38f4-4842-b1ac-1e69b0045607%22:%91%22header_2_text_color%22%93}”]<\/p>\n

Geometric Progression<\/strong><\/h2>\n

Consider a series with the terms 2, 4, 8, 16, 32, and so on. Now consider another series that contains the terms 6, 18, 54, 162, and so on. You might note that the ratio of the second and the first number or the third and the second number in the above series are the same. In series one, the common ratio is 2 while in series two, the common ratio is 3. The series in which<\/strong> the ratio between any two adjacent numbers is the same is known as geometric progression.<\/strong><\/p>\n

So, a geometric progression can be expressed in the form of a, ar, ar2<\/sup>, ar3<\/sup>, ar4<\/sup>\u2026\u2026arn-1<\/sup><\/p>\n

Where a = first number in the series
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 r = common ratio<\/p>\n

A geometric progression that has an infinite number of terms in the series is known as an infinite geometric series.<\/p>\n

Sum of geometric progression:<\/strong><\/h2>\n

To understand the sum of an infinite geometric progression, let us first understand the formula and the derivation of the sum of a finite geometric progression.<\/p>\n

As mentioned above, a Geometric Progression having \u2018n\u2019 number of terms can be expressed as a, ar, ar2<\/sup>, ar3<\/sup>, ar4<\/sup>\u2026\u2026arn-1<\/sup><\/p>\n

And the sum of this geometric progression is S =\u00a0 a+ar+ar2<\/sup>+ ar3<\/sup>+ ar4<\/sup>\u2026\u2026+arn-1<\/sup>\u00a0\u00a0\u00a0\u00a0 – Eqn (1)<\/p>\n

Multiply Equation (1) by the common ratio r, we get<\/p>\n

 <\/p>\n

Sr = ar+ar2<\/sup>+ ar3<\/sup>+ ar4<\/sup>\u2026\u2026+arn<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 – Eqn (2)<\/p>\n

Subtracting Eqn (2) from Eqn (1), we get<\/p>\n

(1-r) S = a (1- rn<\/sup>)<\/p>\n

S = (a (1- rn<\/sup> ))\/(1-r) ( where r <1)<\/p>\n

When r >1, S = (a ( rn<\/sup>-1 ))\/(r-1)<\/p>\n

The formula to calculate the sum of an infinite geometric progression is given as S\u221e<\/sub> = a\/(1-r)where r\u22600 and | r | <1<\/p>\n

I.e. The sum of an infinite geometric progression is possible to calculate only when the common ratio is between -1 and 1 and is not 0. If the common ratio is beyond the given range, then the sum tends towards infinity.<\/p>\n

We know that when r <1, the sum of the geometric progression is S =(a (1- rn<\/sup>\u00a0 ))\/(1-r).<\/p>\n

In a geometric progression where n \u2192\u221e,<\/p>\n

S = (a (1- rn<\/sup>))\/(1-r)<\/p>\n

I.e S = (a )\/(1-r)- (a\u00a0 rn<\/sup>)\/(1-r)<\/p>\n

Where n \u2192 \u221e, the value of (a\u00a0 rn<\/sup>)\/(1-r)\u2192 0 when r\u22600 and | r | <1.<\/p>\n

Thus, S = (a )\/(1-r)<\/strong><\/p>\n

 <\/p>\n

Solved Examples<\/strong><\/h3>\n
    \n
  1. Calculate the sum of the first 7 numbers of the geometric series 27, 81, 243\u2026..<\/li>\n<\/ol>\n

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

    In the given geometric progression, the first term \u2018a\u2019 = 27 and the common ratio \u2018r\u2019 is 81\/27= \u00a0<\/p>\n

    We know that when r>1, S = (a ( rn<\/sup>-1 ))\/(r-1)<\/p>\n

    S = (27 x (37<\/sup>-1))\/(3-1)<\/p>\n

    S = 29,511<\/p>\n

     <\/p>\n

      \n
    1. Calculate the sum of an infinite geometric progression having the numbers 32,16,8,4\u2026.<\/li>\n<\/ol>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Solution:<\/strong><\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>The first term \u2018a\u2019 in the infinite series is 32 and the common ratio \u2018r\u2019 is 16\/32= 1\/2<\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 We know the sum of an infinite geometric progression when the common ratio r <1 is<\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 S=\u00a0 (a )\/(1-r)<\/strong><\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 S = \u00a0\u00a032\/(1-0.5)<\/strong>\u00a0<\/strong><\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>s= (32 )\/0.5<\/p>\n

      S = 64<\/p>\n

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      Explore Other Topics<\/h1>\n

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      \n
      Geometry<\/a><\/div>\n
      Trigonometry<\/a><\/div>\n
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      Numbers<\/a><\/div>\n<\/div>\n

      [\/et_pb_text][et_pb_text admin_label=”Related Concepts
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      Related Concepts<\/h1>\n

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      \n
      GP Formula<\/a><\/div>\n
      Sum of a GP<\/a><\/div>\n
      n^{\\text{th}}<\/span> term of a GP<\/a><\/div>\n<\/div>\n

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      Frequently Asked Questions<\/h1>\n
        \n
      1. What is an infinite geometric progression?<\/li>\n<\/ol>\n

        A geometric progression that has an infinite number of terms is known as an infinite geometric progression<\/p>\n

          \n
        1. What is the formula for the sum of an infinite geometric progression?<\/li>\n<\/ol>\n

          The formula for calculating the sum of an infinite geometric progression is S=\u00a0 (a )\/(1-r)<\/strong> where a is the first term and r is the common ratio, and \u00a0\u00a0r\u22600 and | r | <1<\/p>\n

            \n
          1. Is it possible to calculate the sum of an infinite geometric progression where the common ratio is greater than 1?<\/li>\n<\/ol>\n

            The sum of an infinite geometric progression can be calculated only when the common ratio is between -1 and 1 and \u22600. If the common ratio is not within -1 and 1, then the sum of the infinite series will tend towards infinity.<\/p>\n

            [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

            Sum of infinite geometric progressionGeometric Progression Consider a series with the terms 2, 4, 8, 16, 32, and so on. Now consider another series that contains the terms 6, 18, 54, 162, and so on. You might note that the ratio of the second and the first number or the third and the second number […]<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":714,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"yoast_head":"\nSum of infinite geometric progression - mydomain<\/title>\n<meta name=\"description\" content=\"The formula to calculate the sum of an infinite geometric progression is S\u221e =a\/(1-r), where a = first number of the series and r = common ratio\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/sum-of-infinite-geometric-progression\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sum of infinite geometric progression - 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