{"id":2496,"date":"2021-10-01T07:09:26","date_gmt":"2021-10-01T07:09:26","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=2496"},"modified":"2022-01-02T06:30:09","modified_gmt":"2022-01-02T06:30:09","slug":"nth-term-of-a-geometric-progression","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/nth-term-of-a-geometric-progression\/","title":{"rendered":"nth term of a Geometric Progression"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” 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” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|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=”4.10.8″ _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

nth term of a Geometric Progression<\/strong><\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.11.3″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” header_3_text_color=”#777777″ custom_padding=”15px|15px||4px|false|false” global_colors_info=”{}”]<\/p>\n

nth<\/sup>\u00a0term of a Geometric Progression<\/strong><\/h2>\n

The series in which the ratio between any two consecutive numbers is the same is known as geometric progression. For example, consider a series containing the terms 1, 2, 4, 8, 16, and so on. The ratio of any two consecutive numbers i.e \\frac{2}{1} = \\frac{4}{2} = \\frac{16}{8} = 2<\/span><\/span><\/p>\n

So, a geometric progression can be expressed in the form of a, ar, ar\u00b2, ar\u00b3, ar4<\/sup>\u2026\u2026arn-1<\/sup><\/span><\/p>\n

Where a = the first term<\/p>\n

r = common ratio<\/p>\n

Now let us learn how to find the nth<\/sup><\/span> term in a geometric progression.<\/p>\n

Formula for the nth<\/sup>\u00a0term of a Geometric Progression<\/strong><\/h2>\n

Consider a geometric progression having the terms 5, 15, 45, 135, 405, and 1215.<\/p>\n

The first term is 5 and the common ratio of the above series is 3.<\/p>\n

 <\/p>\n

Now, we can write the different terms in the above series as below.<\/p>\n

T1 <\/sub>= 5\u00a0 = a<\/p>\n

T2<\/sub> = 15 = 5X3 = ar<\/p>\n

T3<\/sub> = 45 = 5X32<\/sup> = ar2<\/sup><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/sup>T4<\/sub> = 135 = 5X33<\/sup> = ar3<\/sup><\/p>\n

 <\/p>\n

Can you spot the pattern here? The pattern that is emerging is that the nth<\/sup><\/span> term in a geometric progression is the product of the first term and the common ratio raised to (n-1).<\/p>\n

 <\/p>\n

So, the formula to calculate the nth<\/sup><\/span>\u00a0<\/sup>term of a GP is Tn<\/sub> = arn-1<\/sup><\/p>\n

 <\/p>\n

<\/h2>\n

<\/h2>\n

Solved Examples:<\/strong><\/h3>\n
    \n
  1. If 5, 20, 80, 320\u2026 is a geometric progression, find the 7th<\/sup><\/span> term of the series.<\/li>\n<\/ol>\n

     <\/p>\n

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

    From the given question, we know that the first term a = 5 and the common ratio r =20\/5=4<\/p>\n

    The formula to calculate the nth<\/sup><\/span> term of a geometric progression is Tn<\/sub> = arn-1<\/sup><\/p>\n

    T7<\/sub>\u00a0 =5 \u2715 46<\/sup><\/p>\n

    T7<\/sub>\u00a0 =5 \u2715 4,096<\/p>\n

    T\u00a0 =20,480<\/p>\n

      \n
    1. If the 3rd<\/sup> term and 5th<\/sup> term of a GP are 40 and 160, find the 8th<\/sup> term of the GP.<\/li>\n<\/ol>\n

       <\/p>\n

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

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>We know that the nth<\/sup> term of a geometric progression can be expressed as Tn<\/sub> = arn-1<\/sup><\/p>\n

       <\/p>\n

      So, T3<\/sub> = 40 =\u00a0 ar3-1<\/sup> and<\/p>\n

      T5<\/sub> = 160 = ar5-1<\/sup><\/p>\n

      40 = ar2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1)<\/p>\n

      160 = ar4<\/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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2)<\/p>\n

      Now, dividing (2) by (1), we get<\/p>\n

      4 = r2<\/sup><\/p>\n

      r = \u221a4<\/p>\n

      r = 2.<\/p>\n

      Substituting r = 2 in (1), we get<\/p>\n

      40 = ax22<\/sup><\/p>\n

      a = 10.<\/p>\n

      To find the 8th<\/sup> term of the GP, we can use the formula\u00a0 Tn<\/sub> = arn-1<\/sup><\/p>\n

      T8<\/sub> = 10 x 28-1<\/sup><\/p>\n

      `T8<\/sub> = 10 x 128<\/p>\n

      T8<\/sub> = 1280.<\/p>\n

       <\/p>\n

        \n
      1. How many terms are there in a GP containing the terms 3, 6, 12, \u2026.. 192?<\/li>\n<\/ol>\n

         <\/p>\n

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

        In the given question, a= 3 and r =6\/3=2. \u00a0and the last term is 192.<\/p>\n

        We know the nth<\/sup><\/span> term of a GP is Tn<\/sub> = arn-1<\/sup><\/p>\n

        In the given question, Tn<\/sub> = 192 = 3 x 2n-1<\/sup><\/p>\n

        64= 2n-1<\/sup> = 26<\/sup><\/p>\n

        n = 6+1 = 7.<\/p>\n

        Hence, there are 7 terms in the given GP series.<\/p>\n

         <\/p>\n

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        Explore Other Topics<\/h1>\n

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        \n
        Geometry<\/a><\/div>\n
        Trigonometry<\/a><\/div>\n
        Operations<\/a><\/div>\n
        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
        Geometric Progression<\/a><\/div>\n
        AP and GP<\/a><\/div>\n
        Sum of infinite GP<\/a><\/a><\/div>\n<\/div>\n

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        Frequently Asked Questions\u00a0<\/span><\/h1>\n

        1. How to calculate the sum of a GP series?<\/strong><\/p>\n

        Ans<\/strong> We can calculate the sum of a geometric progression series using the following formulas:<\/p>\n

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

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

         <\/p>\n

          \n
        1. How to calculate the sum of an infinite GP series?<\/strong><\/li>\n<\/ol>\n

          Ans<\/strong> The sum of an infinite GP series is calculated using the formula S = (a (1- r^n\u00a0 ))\/(1-r) \u00a0where |r| <1 .<\/p>\n

           <\/p>\n

            \n
          1. How to find the value of r in a GP?<\/strong><\/li>\n<\/ol>\n

            Ans<\/strong>: r is the common ratio of any two consecutive numbers in a GP series. The value of r is constant in any given geometric progression. The value of r can be calculated using the formula Tn\/(Tn-1). We can also obtain the value of r by dividing any term in the GP by its preceding term.<\/p>\n

             <\/p>\n

              \n
            1. How to calculate the number of terms in a GP?<\/strong><\/li>\n<\/ol>\n

              Ans<\/strong>: Where the first term, common ratio, and the last term of a geometric progression are given, we can find the number of terms in the given GP by substituting the given values in the formula\u00a0 Tn<\/sub> = arn-1<\/sup> and solving for n.<\/p>\n

              \u200b<\/p>\n

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

              Numbers can be expressed in words and we call them number names. The numbers starting from 1 to 20 are written with specific spellings followed by a generic pattern. The nth term of a Geometric Progression is Tn = arn-1. Common ratio = r = TnTn-1. The general form of a GP is a, ar, ar2, ar3, ar4\u2026\u2026arn-1<\/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":"\nnth term of a Geometric Progression - mydomain<\/title>\n<meta name=\"description\" content=\"The nth term of a Geometric Progression is Tn = arn-1. Common ratio = r = Tn\/(Tn-1). 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