{"id":2835,"date":"2021-10-08T06:25:56","date_gmt":"2021-10-08T06:25:56","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=2835"},"modified":"2022-01-02T13:47:34","modified_gmt":"2022-01-02T13:47:34","slug":"sum-of-odd-numbers-explanation-and-examples","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/sum-of-odd-numbers-explanation-and-examples\/","title":{"rendered":"Sum of odd numbers \u2013 Explanation and Examples"},"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
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Sum of odd numbers \u2013 Explanation and Examples<\/strong><\/h1>\n

[\/et_pb_text][et_pb_text _builder_version=”4.11.3″ _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

What is an odd natural number? <\/strong><\/h2>\n

 <\/p>\n

A system of natural numbers that is not a multiple of 2 is known as a set of odd natural numbers. For example, 1, 3, 5, 7, \u2026. are odd natural numbers because upon division by 2, they give a fractional form. Odd numbers are placed alternatively in a series of natural numbers, and by using an arithmetic progression, we can calculate the sum of odd natural numbers very quickly.<\/p>\n

What is the formula to find the sum of odd numbers?<\/strong><\/h2>\n

 <\/p>\n

Now that you recognize odd numbers, we will derive the formula for the sum of first n odd numbers by arithmetic progressions. Arithmetic Progression (A.P) is a sequence of numbers in which the difference between any two consecutive numbers remains constant. So, for example, the series of odd natural numbers, i.e., 1, 3, 5, 7, 9, \u2026. is an A.P because the difference between two consecutive odd numbers remains 2.<\/p>\n

Here is how AP can help us derive the answer.<\/p>\n

Let the sum of first n odd numbers be Sn<\/sub><\/p>\n

Sn <\/sub>= 1 + 3 + 5 + 7 + 9 + \u2026\u2026\u2026\u2026\u2026\u2026\u2026.. + (2n-1)\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– equation 1<\/p>\n

Wondering how we found the last term of this AP series?<\/p>\n

According to the above series, a = 1 and d = 2<\/p>\n

The formula for finding the nth<\/sup> term of an AP is a + (n – 1)d<\/p>\n

Therefore, upon putting these values in the nth<\/sup> formula, we get the last term as (2n – 1)<\/p>\n

By arithmetic progression, we know the formula to find the sum of n numbers;<\/p>\n

Sn<\/sub> = \u00bd x n [2a + (n-1) d]\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 – equation 2<\/p>\n

 <\/p>\n

In this equation,<\/p>\n

n = total number of digits in the series<\/p>\n

a = first digit of the A.P<\/p>\n

d = common difference in the A.P<\/p>\n

With respect to equation 1, we know that;<\/p>\n

a = 1, d= 2<\/p>\n

 <\/p>\n

Upon substituting these values in equation 2 with respect to equation 1;<\/p>\n

Sn<\/sub> = (n\/2) x (a + l)<\/p>\n

Sn<\/sub> = (n\/2) x (1 + 2n \u2013 1)<\/p>\n

Sn<\/sub> = (n\/2) x (2n)<\/p>\n

Sn<\/sub> = n2<\/sup><\/p>\n

Therefore, sum of first n odd numbers (Sn<\/sub>) = n2<\/sup><\/p>\n

\u00a0<\/sup><\/p>\n

Solved examples<\/strong><\/h2>\n

 <\/p>\n

Question 1: How many odd numbers are there between 1 to 100? Find their sum.<\/p>\n

Solution:<\/p>\n

We know that there is a total of 50 odd numbers between 1 and 100. According to the formula for the sum of first n odd numbers Sn<\/sub> = n2<\/sup><\/p>\n

Here, n = 50<\/p>\n

S50<\/sub> = (50)2<\/sup><\/p>\n

S50<\/sub> = 2500<\/p>\n

Hence, the sum of odd numbers between 1 and 100 is 2500.<\/p>\n

 <\/p>\n

Question 2: Prove that the formula and sum of first n natural odd numbers manually give the same answer.<\/p>\n

Solution:<\/p>\n

Starting from 1, suppose we have n odd natural numbers, 1, 3, 5, 7,\u2026\u2026 (2n-1)<\/p>\n

So,<\/p>\n

The sum of the first odd natural number is 1<\/p>\n

Sum of first two odd natural numbers = 1 + 3 = 4 = 22<\/sup><\/p>\n

Sum of first three odd natural numbers = 1 + 3 + 5 = 9 = 32<\/sup><\/p>\n

Sum of first four odd natural numbers = 1 + 3 + 5 + 7 = 16 = 42<\/sup><\/p>\n

Hence, the formula and sum of first n natural odd numbers manually give the same answer.<\/p>\n

Question 3: Find the odd numbers between 12 and 24 and find their sum.<\/p>\n

Solution:<\/p>\n

The series of odd numbers between 12 and 24 is 13, 15, 17, 19, 21, 23<\/p>\n

here, a = 13, n = 6, d = 2<\/p>\n

Putting these values in Sn<\/sub> = \u00bd x n [2a + (n-1) d]<\/p>\n

S6<\/sub> = \u00bd x 6 [2 x 13 + (6 -1) 2]<\/p>\n

S6<\/sub> = \u00bd x 6 [26 + 10]<\/p>\n

S6<\/sub> = 3 x 36<\/p>\n

S6<\/sub> = 108<\/p>\n

So, sum of the series 13, 15, 17, 19, 21, 23 is 108.[\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics
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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
Arithmetic Progression<\/a><\/div>\n
Sum of AP<\/a><\/div>\n
Sum of an Infinite AP<\/a><\/div>\n<\/div>\n

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Frequently Asked Questions<\/h1>\n
    \n
  1. What are some properties of odd numbers?<\/li>\n<\/ol>\n
      \n
    • The sum of two odd numbers gives an even number<\/li>\n
    • The subtraction of two odd numbers gives an even number<\/li>\n
    • The multiplication of two odd numbers gives an even number<\/li>\n
    • The division of two odd numbers gives an odd number<\/li>\n<\/ul>\n
        \n
      1. Can we categorize fractions and decimals as odd numbers?<\/li>\n<\/ol>\n

        No, fractions and decimals are neither odd nor even.<\/p>\n

          \n
        1. What is the formula to find the sum of n odd natural numbers?<\/li>\n<\/ol>\n

          The formula for finding the sum of n odd natural number is Sn<\/sub> = \u00bd x n [2a + (n-1) d].<\/p>\n

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

          Sum of odd numbers \u2013 Explanation and ExamplesWhat is an odd natural number?   A system of natural numbers that is not a multiple of 2 is known as a set of odd natural numbers. For example, 1, 3, 5, 7, \u2026. are odd natural numbers because upon division by 2, they give a fractional […]<\/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 odd numbers \u2013 Explanation and Examples - mydomain<\/title>\n<meta name=\"description\" content=\"The sum of odd numbers is given by the equation Sn = n2, where n is the total numbers in the series. 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