[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
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Any number in the form of (2n + 1), where n is any positive integer is termed as an odd number. 2 is not a factor of these numbers.<\/span><\/p>\n Also for negative odd numbers, it can be written as (2n – 1) where n is any negative integer.<\/span><\/p>\n The series of odd numbers in which every number is followed by its successive odd number are known as consecutive odd numbers. The series of odd numbers are arranged in ascending order.<\/span><\/p>\n The difference between any two consecutive numbers of this series is equal to 2.<\/span><\/p>\n Examples<\/strong><\/p>\n 17, 21 and 23 are not odd consecutive numbers because there is an odd number 19 in between 17 and 21.<\/span><\/p>\n If the first number in this series is of the form (2a + 1), the next numbers of these series are (2a + 3), (2a + 5), ( 2a + 7),( 2a + 9) and so on. Here \u201ca\u201d is a positive integer.<\/span><\/p>\n If the first number in this series is of the form (2a – 1), the next numbers of these series are (2a + 1), (2a + 3), (2a + 5), ( 2a + 7), ( 2a + 9) and so on. Here \u201ca\u201d is a negative integer.<\/span><\/p>\n If there are an odd number of terms in the series of odd consecutive numbers, then the sum of all the terms is odd.<\/span><\/p>\n If there are an even number of terms in the series of odd consecutive numbers, then the sum of all the terms is even.<\/span><\/p>\n <\/p>\n <\/b><\/p>\n <\/span><\/p>\n For obtaining the next 5 numbers of the series we have to keep adding 2 to the previous<\/span><\/p>\n (-51) + 2 = (-49)<\/span><\/p>\n (-49) + 2 = (-47)<\/span><\/p>\n (-47) + 2 = (-45)<\/span><\/p>\n (-45) + 2 = (-43)<\/span><\/p>\n (-43) + 2 = (-41)<\/span><\/p>\n Hence the next five numbers of the given series are -49, -47, -45, -43 and -41.<\/span><\/p>\n <\/p>\n <\/span><\/p>\n Let one of the numbers be (2a + 1).\u00a0<\/span><\/p>\n Next odd number = ( 2a + 1) +2 = (2a + 3)<\/span><\/p>\n Sum = 92(Given)<\/span><\/p>\n \u21d2 (2a + 1) + (2a + 3) = 92<\/span><\/p>\n \u21d2 4a + 4 = 92<\/span><\/p>\n \u21d2 4a = 92 \u2013 4 = 88<\/span><\/p>\n \u21d2 a = \\frac{88}{4}<\/span> = 22<\/span><\/p>\n \u2234 2a + 1 = (2 \u00d7 22) + 1 = 45<\/span><\/p>\n And, 2a + 3 = (2 \u00d7 22) + 3 = 47<\/span><\/p>\n Check\u00a0<\/span><\/p>\n 45 + 47 = 92<\/span><\/p>\n Hence the two numbers are 44 and 47.<\/span><\/p>\n <\/p>\n <\/span><\/p>\n Let one of the numbers be (2a + 1).<\/span><\/p>\n Next odd number = (2a + 1) + 2 = (2a + 3).<\/span><\/p>\n <\/span><\/p>\n Product = (2a+1) (2a+3)<\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= 4 a^{2}+8 a+3<\/span><\/p>\n <\/span><\/p>\n Now, Product = 63(Given)<\/span><\/p>\n \\Rightarrow \\quad 4 a^{2}+8 a+3=63<\/span><\/span><\/p>\n \\Rightarrow \\quad 4 a^{2}+8 a=63-3=60 <\/span> 2a + 1 = (2 \u00d7 3) + 1\u00a0<\/span><\/p>\n \u21d2 2a + 1 = 7<\/span><\/p>\n 2a + 3 = (2 \u00d7 3) + 3\u00a0<\/span><\/p>\n \u21d2 2a + 3 = 9<\/span><\/p>\n Check<\/span><\/p>\n 7 \u00d7 9 = 63<\/span><\/p>\n Therefore the required two numbers are 7 and 9.<\/span><\/p>\n <\/span><\/p>\n [\/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 [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n\n
Solved Examples<\/b><\/h2>\n
1. Given below is a series of odd numbers-<\/h3>\n
-55, -53, -51 and so on. Find the next 5 numbers of this series.<\/h3>\n
2. The sum of two odd numbers is equal to 92. The difference between these numbers is equal to 2. Find them.<\/h3>\n
3. The product of two odd numbers is equal to 63. They are consecutive. Find them.<\/h3>\n
\\Rightarrow \\quad 4\\left(a^{2}+2 a\\right)=60 <\/span>
\\Rightarrow \\quad\\left(a^{2}+2 a\\right)=\\frac{60} {4}=15 <\/span>
\\Rightarrow \\quad a(a+2)=15 <\/span>
\\Rightarrow \\quad a=3<\/span><\/p>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n