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An arithmetic progression is a series, where the difference between every two successive terms in the series is constant. For example, consider a series 7, 11, 15, 19, 23 and 27. In this series, the difference between every two successive terms is 4 and hence this is an arithmetic progression. This is a series containing 6 terms. If an arithmetic progression series has an infinite number of terms, it is called an infinite arithmetic progression. Let us learn how to calculate the sum of an infinite arithmetic progression here.<\/p>\n
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Before learning the sum of an infinite AP, it is important to understand the concept of divergent and convergent.<\/span><\/p>\n A series is said to be a <\/span>convergent series<\/b> if its sum approaches a finite number. A good example of a convergent series is an infinite geometric progression where the |r| < 1 and r\u2260 0 where r is the common ratio of the GP.\u00a0<\/span><\/p>\n A <\/span>divergent series<\/b> is an infinite series that is not convergent. An infinite series where the numbers do not approach zero is diverging. An infinite arithmetic progression is an example of a diverging series.\u00a0<\/span><\/p>\n In an infinite arithmetic progression where n is the number of terms, <\/span>n \u2192 \u221e , <\/span>and the common difference is greater than 0, the sum of the arithmetic progression would be infinitely large, and S<\/span>\u221e<\/span> = \u221e.<\/span><\/p>\n Similarly, in an infinite arithmetic progression where <\/span>n \u2192 \u221e <\/span>and has a common difference less than 0, then the terms of the Arithmetic Progression are approaching -\u221e and the sum of such series would be S<\/span>\u221e<\/span> = -\u221e.<\/span><\/p>\n For example, the sum of an infinite series having the terms -2, 0, 2, 4, 6\u2026…\u221e will be \u221e since the common difference 2 is greater than 0. The sum of infinite series having the terms 7, 3, -1, -5 \u2026.. -\u221e will be -\u221e since the common difference (-4) < 0.<\/span><\/p>\n <\/p>\n Solution:<\/b><\/p>\n From the given question, we can form a series having the terms 100, 120, 140, \u2026, 380 where the first term a = 100, common difference d = 20, and the number of terms n = 15. We can calculate the total sum of amount Mr. Y has on Day 15 by using the formula: S_{n}=\\frac{n}{2}[2 a+(n-1) d]<\/span><\/span><\/b><\/p>\n Substituting the given figures in the above formula, we have:<\/span><\/p>\n S_{15}=\\frac{15}{2}[2(100)+(15-1) 20]<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0=\\frac{15}{2}[200+(14) 20]<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0=\\frac{15}{2}[200+280]<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0=\\frac{15}{2}\\times 480<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0=15\\times 240<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0=3600<\/span><\/span><\/p>\n \\therefore S_{15}=3600<\/span><\/span><\/p>\n Hence, the total amount Mr. Y had on Day 15 was Rs. 3600.<\/span><\/p>\n 2. Find the sum of an infinite AP having the terms 1, 5, 9, 13, …., \u221e.<\/span><\/p>\n Since infinite arithmetic progression series are divergent series, the sum of an infinite arithmetic series can not exactly be determined. The common difference in this AP is 4 and so we know that the sum of the given AP is \u221e.<\/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>\nSolved Examples:<\/strong><\/h2>\n
1. If Mr. Y had Rs. 100 on Day 1 and every succeeding day, he received Rs. 20 more than what he received the previous day, for 15^{\\text {th}}<\/span>days. How much money does he have at the end of the <\/span>15^{\\text {th}}<\/span> <\/span>day?<\/span><\/h3>\n
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\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