{"id":2454,"date":"2021-10-01T04:26:08","date_gmt":"2021-10-01T04:26:08","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=2454"},"modified":"2022-01-02T06:21:56","modified_gmt":"2022-01-02T06:21:56","slug":"arithmetic-progression","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/arithmetic-progression\/","title":{"rendered":"Arithmetic Progression"},"content":{"rendered":"

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Arithmetic Progression<\/strong><\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.11.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” header_3_text_color=”#898989″ custom_padding=”15px|15px||4px|false|false” global_colors_info=”{}”]<\/p>\n

Arithmetic Progression – Definition and formulas<\/strong><\/h2>\n

An arithmetic progression (AP) is a sequence, where the difference between any two consecutive numbers is constant. For example, consider the 5 tables. We have 5, 10, 15, 20, and so on. The difference between any two successive numbers in this series is always 5. So this is an arithmetic progression with the common difference of 5.\u00a0<\/span><\/p>\n

We can find arithmetic progression series all around us. We can find AP in a taxi fare where the fare per additional kilometer increases by a fixed amount or the fixed percentage of interest in a fixed deposit. Let us learn in detail about the various terms and formulas in arithmetic progressions here.\u00a0<\/span><\/p>\n

<\/h2>\n

Terms in an Arithmetic Progression:<\/strong><\/h2>\n

An arithmetic progression can be specified by two terms which are the first term, and the common difference.<\/span><\/p>\n

The first term, as the name suggests, is the first term of the arithmetic sequence and is denoted generally as a. In an AP containing the terms 7, 11, 15, 19, 23 \u2026, the first term a = 7.\u00a0<\/span><\/p>\n

The common difference is the constant number which is the difference between any two consecutive terms in an AP. We can add the common difference to the previous term to arrive at the next term in the series. It is generally denoted as d.\u00a0<\/span><\/p>\n

Common difference d =<\/span>\u00a0an <\/sub>– an-1<\/sub><\/p>\n

Where<\/span><\/p>\n

d\u00a0 \u00a0 = common difference<\/span><\/p>\n

an<\/sub> <\/span>= nth term in an Arithmetic Progression<\/span><\/p>\n

an-1 <\/sub>= (n-1)th term in an Arithmetic Progression<\/span><\/p>\n

In the above mentioned series, the common difference d is (11-7) = (15-11) = 4<\/span><\/p>\n

The general form of an arithmetic progression is a, a+d, a+2d, a+3d, \u2026.. a+(n-1)d<\/span><\/p>\n

Where<\/span><\/p>\n

a= first term<\/span><\/p>\n

d= common difference and<\/span><\/p>\n

n= number of terms<\/span><\/p>\n

<\/h2>\n

Formulas in Arithmetic Progression:<\/b>\u00a0<\/span><\/h2>\n

<\/h2>\n

nth term of an AP\u00a0<\/strong><\/p>\n

In an arithmetic progression where the first term and the common difference is known, the nth term of the series can be calculated using the formula Tn <\/sub><\/span>= ((a+(n-1))d<\/span>\u00a0<\/span><\/p>\n

\u00a0If we want to calculate the nth term from the end of the AP, then we use the formula\u00a0 <\/span>
<\/span>T<\/span>n<\/sub>=( l-(n-1))d<\/span>\u00a0<\/span><\/p>\n

where <\/span> l<\/span> is the last term in an AP<\/span><\/p>\n

Number of terms in an AP<\/strong><\/h3>\n

When the first and last terms of an AP and the common difference is known, then we can calculate the number of terms in the arithmetic progression using the formula <\/span>n=<\/span>(l-a)<\/span>d<\/span>+1<\/span><\/p>\n

Sum of first n terms in an AP<\/strong><\/h3>\n

When we know the first term and the common difference in the AP, the formula for calculating the sum of an arithmetic progression is S=<\/span> n[2a+(n-1)d]\/<\/span>2<\/span>.<\/span><\/p>\n

When we know the first and the last term, the sum of n terms in an AP is <\/span>S=<\/span>n(a+l)\/<\/span>2<\/span><\/p>\n

Where a is the first term and l is the last term.<\/span><\/p>\n

Sum of an infinite arithmetic progression<\/strong><\/h3>\n

If the common difference in the AP is greater than 0, the sum of the AP tends to +\u221e , and the sum of an AP where the common difference is less than 0, tends to -\u221e .<\/span><\/p>\n

<\/h2>\n

Formula summary<\/strong><\/h2>\n

Where\u00a0<\/span><\/p>\n

a= first term<\/span><\/p>\n

d= common difference and<\/span><\/p>\n

n= number of terms<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span> l<\/span>= last term in an AP<\/span><\/p>\n

<\/h2>\n\n\n\n\n\n\n\n\n\n
Nth term of an AP<\/span><\/td>\nT<\/span>n <\/span>= (a+(n-1))\/d<\/span><\/td>\n<\/tr>\n
Nth term from the end of an AP<\/span><\/td>\nT<\/span>n <\/span>=( l-(n-1))\/d<\/span>\u00a0<\/span><\/td>\n<\/tr>\n
Number of terms in an AP<\/span><\/td>\nn=<\/span>(l-a)\/<\/span>d<\/span>+1<\/span><\/td>\n<\/tr>\n
Sum of n terms in an AP<\/span><\/td>\n S=<\/span> n[2a+(n-1)d]\/<\/span>2<\/span><\/td>\n<\/tr>\n
Sum of n terms in an AP when first and last terms are known<\/span><\/td>\nS=<\/span>n(a+l)\/<\/span>2<\/span><\/td>\n<\/tr>\n
Sum of infinite AP when common difference >0<\/span><\/td>\nS\u2192+\u221e<\/span><\/td>\n<\/tr>\n
Sum of infinite AP when common difference < 0<\/span><\/td>\nS\u2192-\u221e<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/h2>\n

Solved Examples:<\/strong><\/h3>\n
    \n
  1. Find the sum of the natural numbers from 501 to 1000.\u00a0<\/span><\/li>\n<\/ol>\n

    We can calculate the sum\u00a0 of the numbers in an Arithmetic Progression where the first and the last terms are known using the formula <\/span>S= <\/span>n(a+l))\/<\/span>2<\/span><\/p>\n

    We need to find the sum of the natural numbers from 501 to 1000, where the first term is 501, the last term is 1000 and the common difference is 1. The number of terms in this arithmetic progression is 500.<\/span><\/p>\n

    Substituting the above figures in the formula, we get<\/span><\/p>\n

    S<\/span>50<\/span>=<\/span>n(a+l)\/<\/span>2<\/span><\/p>\n

    S<\/span>50<\/span>=<\/span>500(501+1000)\/<\/span>2<\/span><\/p>\n

    S<\/span>50<\/span>=<\/span>7,50,500<\/span>2<\/span><\/p>\n

    S<\/span>50<\/span>=3,75,250<\/span><\/p>\n

    The sum of the natural numbers from 501 to 1000 is 3,75,250.<\/span><\/p>\n

    2. In a building, the first floor has 69 rooms, the second floor has 64 rooms, the third floor has 59 rooms, and so on. The final floor has 9 rooms. Find the total number of floors in the building.\u00a0<\/span><\/p>\n

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

    The number of rooms in this building is 69, 64, 59, \u2026.. 9. We can see that this forms an<\/span> arithmetic progression with the first term a = 69, common difference d = 64-69 = -5 and the n<\/span>th<\/span> term is 9.\u00a0<\/span><\/p>\n

    We know that the n<\/span>th<\/span> term of an AP a<\/span>n<\/span> = a+ (n-1)\/d<\/span><\/p>\n

    Substituting the given values in this formula, we get<\/span><\/p>\n

    \u00a0 9 = 69 + (n-1)(-5)<\/span><\/p>\n

    -60 = -5n + 5.<\/span><\/p>\n

    65\u00a0 = 5n<\/span><\/p>\n

    \u00a0 n\u00a0 = 13.<\/span><\/p>\n

    So, the building has 13 floors.<\/span><\/p>\n

    3. Find the sum of the first 20 terms of the AP 4, 11, 18, 25 \u2026.<\/span><\/p>\n

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

    \u00a0This is an AP with the first term a = 4 and the common difference d = 11-4 = 7.<\/span><\/p>\n

    The sum of first n terms of an AP S<\/span>n <\/span>=<\/span> n[2a+(n-1)d]\/<\/span>2<\/span><\/p>\n

    S<\/span>20 <\/span>=<\/span> 20[2\u27154+(20-1)7]\/<\/span>2<\/span><\/p>\n

    S_{20} =\\frac{20[2*4+(20-1)7]}{2}<\/span><\/span><\/span><\/p>\n

    S<\/span>20 <\/span>=<\/span> 20[141]\/<\/span>2<\/span><\/p>\n

    S<\/span>20 <\/span>=<\/span> 1,410<\/span><\/p>\n

    [\/et_pb_text][et_pb_text disabled_on=”on|on|on” admin_label=”Sample Questions
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    Practice Multiple Choice Questions<\/h1>\n

    [\/et_pb_text][et_pb_text disabled_on=”on|on|on” admin_label=”Question 1″ _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” custom_padding=”25px|25px|25px|25px|true|true” border_radii=”on|15px|15px|15px|15px” border_width_all=”2px” border_color_all=”#000000″ border_width_top=”4px” border_color_top=”#E02B20″ disabled=”on” global_colors_info=”{}”]<\/p>\n

    \n
    \n

    Questions<\/strong><\/p>\n

      \n
    1. Find the sum of the natural numbers from 501 to 1000.\u00a0<\/span><\/li>\n<\/ol>\n

      Solution:<\/strong>
      \nWe can calculate the sum\u00a0 of the numbers in an Arithmetic Progression where the first and the last terms are known using the formula <\/span>S= <\/span>n(a+l))<\/span>2<\/span>
      \n 
      \nWe need to find the sum of the natural numbers from 501 to 1000, where the first term is 501, the last term is 1000 and the common difference is 1. The number of terms in this arithmetic progression is 500.<\/span>
      \n 
      \nSubstituting the above figures in the formula, we get<\/span>
      \n 
      \nS<\/span>50<\/span>=<\/span>n(a+l)<\/span>2<\/span>
      \nS<\/span>50<\/span>=<\/span>500(501+1000)<\/span>2<\/span>
      \nS<\/span>50<\/span>=<\/span>7,50,500<\/span>2<\/span>
      \nS<\/span>50<\/span>=3,75,250<\/span>
      \n 
      \nThe sum of the natural numbers from 501 to 1000 is 3,75,250.<\/span>
      \n\u00a0<\/span>
      \n <\/p>\n

        \n
      1. In a building, the first floor has 69 rooms, the second floor has 64 rooms, the third floor has 59 rooms, and so on. The final floor has 9 rooms. Find the total number of floors in the building.\u00a0<\/span><\/li>\n<\/ol>\n

        Solution:<\/b>
        \n 
        \nThe number of rooms in this building is 69, 64, 59, \u2026.. 9. We can see that this forms an<\/span> arithmetic progression with the first term a = 69, common difference d = 64-69 = -5 and the n<\/span>th<\/span> term is 9.\u00a0<\/span>
        \n 
        \nWe know that the n<\/span>th<\/span> term of an AP a<\/span>n<\/span> = a+ (n-1)d<\/span>
        \nSubstituting the given values in this formula, we get<\/span>
        \n 
        \n \u00a0 9 = 69 + (n-1)(-5)<\/span>
        \n-60 = -5n + 5.<\/span>
        \n65\u00a0 = 5n<\/span>
        \n\u00a0 n\u00a0 = 13.<\/span>
        \nSo, the building has 13 floors.<\/span>
        \n <\/p>\n

          \n
        1. Find the sum of the first 20 terms of the AP 4, 11, 18, 25 \u2026.<\/span><\/li>\n<\/ol>\n

           
          \nSolution:\u00a0<\/b>
          \n\u00a0This is an AP with the first term a = 4 and the common difference d = 11-4 = 7.<\/span>
          \nThe sum of first n terms of an AP S<\/span>n <\/span>=<\/span> n[2a+(n-1)d]<\/span>2<\/span>
          \nS<\/span>20 <\/span>=<\/span> 20[2\u27154+(20-1)7]<\/span>2<\/span>
          \nS<\/span>20 <\/span>=<\/span> 20[141]<\/span>2<\/span>
          \nS<\/span>20 <\/span>=<\/span> 1,410<\/span>\n<\/div>\n<\/div>\n

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          Explore Other Topics<\/h1>\n

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          Related Concepts<\/h1>\n

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          \n
          AP Formula<\/a><\/div>\n
          Sum of an infinite AP<\/a><\/div>\n
          AP and GP<\/a><\/a><\/div>\n<\/div>\n

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

          1.<\/strong> What is an Arithmetic Progression?<\/span><\/p>\n

          Ans:<\/strong> An arithmetic progression is a series, where the difference between any two consecutive numbers is constant.<\/span><\/p>\n

          2.<\/strong> What is the formula to calculate the sum of squares of first n natural numbers?<\/span><\/p>\n

          Ans:<\/strong> We can calculate the sum of squares of the first n natural numbers of an AP is\u00a0<\/span><\/p>\n

          S= <\/span>n(n+1)(2n+1)<\/span>6<\/span><\/p>\n

          3.<\/strong> What is the formula to calculate the sum of first n terms in an AP?<\/span><\/p>\n

          Ans: The formula for calculating the sum of an arithmetic progression is S=<\/span> n[2a+(n-1)d]<\/span>2<\/span>.<\/span><\/p>\n

          4.<\/strong> What are the different progressions in maths?<\/span><\/p>\n

          Ans:<\/strong> There are three different progressions in maths namely arithmetic progression, geometric progression, and harmonic progression.\u00a0<\/span><\/p>\n

          5.<\/strong> What is the sum of an infinite AP?<\/span><\/p>\n

          Ans:<\/strong> If the common difference in the AP is greater than 0, the sum of the AP tends to +\u221e , and the sum of an AP where the common difference is less than 0, tends to -\u221e .<\/span><\/p>\n

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

          An AP is any series that has a common difference between any two consecutive numbers. The sum of the first n terms of the AP can be calculated by using S= (n[2a+(n-1)d])\/2.<\/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":"\nArithmetic Progression - mydomain<\/title>\n<meta name=\"description\" content=\"An AP is any series that has a common difference between any two consecutive numbers. 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