{"id":2482,"date":"2021-10-01T06:34:49","date_gmt":"2021-10-01T06:34:49","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=2482"},"modified":"2022-01-02T06:31:19","modified_gmt":"2022-01-02T06:31:19","slug":"nth-term-of-an-arithmetic-progression","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/nth-term-of-an-arithmetic-progression\/","title":{"rendered":"nth term of an Arithmetic Progression"},"content":{"rendered":"<p>[et_pb_section fb_built=&#8221;1&#8243; admin_label=&#8221;Section&#8221; module_class=&#8221;mainsec&#8221; _builder_version=&#8221;4.10.4&#8243; _module_preset=&#8221;default&#8221; background_color=&#8221;#e0f2fd&#8221; z_index=&#8221;1&#8243; custom_padding=&#8221;5px||5px||true|false&#8221; locked=&#8221;off&#8221; collapsed=&#8221;off&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_row column_structure=&#8221;3_5,2_5&#8243; custom_padding_last_edited=&#8221;on|phone&#8221; _builder_version=&#8221;4.10.8&#8243; _module_preset=&#8221;default&#8221; background_color=&#8221;#FFFFFF&#8221; width=&#8221;100%&#8221; max_width=&#8221;1310px&#8221; custom_padding=&#8221;|51px|40px|51px|false|true&#8221; custom_padding_tablet=&#8221;&#8221; custom_padding_phone=&#8221;|40px|30px|40px|false|true&#8221; border_radii=&#8221;on|10px|10px|10px|10px&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_column type=&#8221;3_5&#8243; admin_label=&#8221;Column L&#8221; _builder_version=&#8221;4.9.10&#8243; _module_preset=&#8221;default&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_text admin_label=&#8221;Acute Angles<br \/>\n&#8221; _builder_version=&#8221;4.10.8&#8243; _module_preset=&#8221;default&#8221; header_font=&#8221;|700|||||||&#8221; header_text_align=&#8221;left&#8221; header_font_size=&#8221;50px&#8221; header_line_height=&#8221;1.18em&#8221; custom_padding=&#8221;|0px||4px|false|false&#8221; header_font_size_tablet=&#8221;&#8221; header_font_size_phone=&#8221;35px&#8221; header_font_size_last_edited=&#8221;on|phone&#8221; global_colors_info=&#8221;{}&#8221;]<\/p>\n<h1><strong>nth term of an Arithmetic Progression<\/strong><\/h1>\n<p>[\/et_pb_text][et_pb_text admin_label=&#8221;Text&#8221; _builder_version=&#8221;4.11.3&#8243; _module_preset=&#8221;default&#8221; text_font_size=&#8221;16px&#8221; header_2_font=&#8221;|600|||||||&#8221; header_2_text_color=&#8221;#a01414&#8243; header_3_font=&#8221;|600|||||||&#8221; header_3_text_color=&#8221;#777777&#8243; custom_padding=&#8221;15px|15px||4px|false|false&#8221; global_colors_info=&#8221;{}&#8221;]<\/p>\n<h2><strong>n<sup style=\"font-size: 10px;\">th<\/sup> term of an Arithmetic Progression:<\/strong><\/h2>\n<p>An arithmetic progression is a series, where the difference between any two consecutive numbers is the same. Consider a series containing the terms 5, 7, 9, 11, and 13. This is an arithmetic progression with a common difference of 2 and the first term is 5. The general form of an arithmetic progression is a, a+d, a+2d, and so on. Let us learn to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term of an arithmetic progression here.<\/p>\n<h2><strong>n<sup style=\"font-size: 10px;\">th<\/sup> term of an Arithmetic Progression Formula:<\/strong><\/h2>\n<p>Consider the series\u00a0 5, 7, 9, 11 and 13. In this series,<\/p>\n<p>T<sub style=\"font-size: 10px;\">1<\/sub> = 5 = a<\/p>\n<p>T<sub style=\"font-size: 10px;\">2<\/sub> = 7 = a+d<\/p>\n<p>T<sub style=\"font-size: 10px;\">3<\/sub> = 9 = a+2d<\/p>\n<p>T<sub style=\"font-size: 10px;\">4 <\/sub>= 11 = a+3d<\/p>\n<p>&nbsp;<\/p>\n<p>We can notice a pattern emerging here. The pattern being, if we need to calculate the 3<sup style=\"font-size: 10px;\">rd<\/sup> term, we need to add the first term with (3-1) times common difference. Similarly, if we need to calculate the 4<sup style=\"font-size: 10px;\">th<\/sup> term, we need to add the first term with (4-1) times the common difference. Thus, to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term in an arithmetic progression, we need to add the first term with (n-1) times the common difference.<\/p>\n<p>&nbsp;<\/p>\n<p>Hence, the formula to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term of an AP is T<sub style=\"font-size: 10px;\">n<\/sub>= a+(n-1)d<\/p>\n<p>&nbsp;<\/p>\n<p>Now let us say we are provided with only the last term of an AP series and we are asked to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term from the end of the AP. To derive the formula for that, let us consider the series 5, 7, 9, 11, and 13 again, only this time, we will consider the series from the end of the series. So, T<sub style=\"font-size: 10px;\">1 <\/sub>will be the last term, T<sub style=\"font-size: 10px;\">2<\/sub> is the second term from the end, T<sub style=\"font-size: 10px;\">3<\/sub> is the third term from the end, and so on. Now we have,<\/p>\n<p>&nbsp;<\/p>\n<p>T<sub style=\"font-size: 10px;\">1<\/sub> = 13 = l<\/p>\n<p>T<sub style=\"font-size: 10px;\">2<\/sub> = 11 = l-d<\/p>\n<p>T<sub style=\"font-size: 10px;\">3<\/sub> = 9 = l-2d<\/p>\n<p>T<sub style=\"font-size: 10px;\">4 <\/sub>= 7 = l-3d<\/p>\n<p>&nbsp;<\/p>\n<p>We can see a pattern emerging here which is, to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term from the end of an AP, we have to deduct (n-1) times the common difference from the last term.<\/p>\n<p>So, the formula to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term from the end of an AP is\u00a0 T<sub style=\"font-size: 10px;\">n<\/sub>= l-(n-1)d<\/p>\n<p>&nbsp;<\/p>\n<p><sub style=\"font-size: 10px;\">\u00a0<\/sub><\/p>\n<h3><strong>Solved Illustrations:<\/strong><\/h3>\n<ol>\n<li>If T<sub style=\"font-size: 10px;\">5<\/sub> = 70 and T<sub style=\"font-size: 10px;\">7<\/sub> = 100, then find T<sub style=\"font-size: 10px;\">9<\/sub> of the AP series<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><strong>Solution:<\/strong><\/p>\n<p>We know that T<sub style=\"font-size: 10px;\">n<\/sub>= a+(n-1)d<\/p>\n<p>So, T<sub style=\"font-size: 10px;\">5<\/sub>=70= a+(5-1)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\u00a0\u00a0\u00a0\u00a0\u00a0 (1)<\/p>\n<p>T<sub style=\"font-size: 10px;\">7<\/sub>=100= a+(7-1)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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2)<\/p>\n<p>Subtracting (2) from (1), we get<\/p>\n<p>30= 2d<\/p>\n<p>d=15<\/p>\n<p>Substituting the value of d in (1), we get<\/p>\n<p>T<sub style=\"font-size: 10px;\">5<\/sub>=70= a+(5-1)15<\/p>\n<p>a=10<\/p>\n<p>Now that we know the first term and the common difference of the AP series, we can calculate the 9<sup style=\"font-size: 10px;\">th<\/sup> term of the AP series as below.<\/p>\n<p>T<sub style=\"font-size: 10px;\">9<\/sub>= 10+(9-1)15<\/p>\n<p>T<sub style=\"font-size: 10px;\">9<\/sub>= 130<\/p>\n<p>&nbsp;<\/p>\n<ol start=\"2\">\n<li>If the n<sup style=\"font-size: 10px;\">th<\/sup> term of an AP is (3n + 1) then find the sum of the first four terms of the series.<\/li>\n<\/ol>\n<p><strong>Solution:<\/strong><\/p>\n<p>The n<sup style=\"font-size: 10px;\">th <\/sup>term of an AP is given as (3n + 1). Substituting 1, 2, 3, and 4 in place of n, we can get the first four terms of the AP as below<\/p>\n<p><strong><\/strong>T<sub style=\"font-size: 10px;\">1<\/sub> = (3(1)+1) = 4<\/p>\n<p>T<sub style=\"font-size: 10px;\">2<\/sub> = (3(2)+1)= 7<\/p>\n<p>T<sub style=\"font-size: 10px;\">3<\/sub> = (3(3)+1)= 10<\/p>\n<p>T<sub style=\"font-size: 10px;\">4 <\/sub>= (3(4)+1) = 13<\/p>\n<p>We know the first four terms of the AP series. The sum of these first four terms is 34.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<ol start=\"3\">\n<li>If the last term of an AP is 86 and the common difference is 7, find the 3<sup style=\"font-size: 10px;\">rd<\/sup> term from the last.<\/li>\n<\/ol>\n<p><strong>Solution:<\/strong><\/p>\n<p>We know that the formula to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term from the end of an AP series is<\/p>\n<p>T<sub style=\"font-size: 10px;\">n<\/sub>= l-(n-1)d.<\/p>\n<p>Substituting the given values, we get<\/p>\n<p>T<sub style=\"font-size: 10px;\">3<\/sub>= 86-(3-1)7<\/p>\n<p>T<sub style=\"font-size: 10px;\">3<\/sub>= 72<\/p>\n<p>\u200b<\/p>\n<p>[\/et_pb_text][\/et_pb_column][et_pb_column type=&#8221;2_5&#8243; module_id=&#8221;stickysideR&#8221; admin_label=&#8221;Column R&#8221; _builder_version=&#8221;4.10.4&#8243; _module_preset=&#8221;default&#8221; background_color=&#8221;#fdefe0&#8243; 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Topics<\/h1>\n<p>[\/et_pb_text][et_pb_text _builder_version=&#8221;4.10.7&#8243; _module_preset=&#8221;default&#8221; text_line_height=&#8221;2.2em&#8221; link_font_size=&#8221;16px&#8221; custom_margin=&#8221;||0px||false|false&#8221; custom_padding=&#8221;10px|15px|10px|28px|true|false&#8221; locked=&#8221;off&#8221; global_colors_info=&#8221;{}&#8221;]<\/p>\n<div>\n<div class=\"trr\"><a href=\"http:\/\/stgwebsite.mindspark.in\/wordpress\/math-concepts\/#geometry\" class=\"otherc\">Geometry<\/a><\/div>\n<div class=\"trr\"><a href=\"http:\/\/stgwebsite.mindspark.in\/wordpress\/math-concepts\/#trigonometry\" class=\"otherc\">Trigonometry<\/a><\/div>\n<div class=\"trr\"><a href=\"http:\/\/stgwebsite.mindspark.in\/wordpress\/math-concepts\/#operations\" class=\"otherc\">Operations<\/a><\/div>\n<div class=\"trr\"><a href=\"http:\/\/stgwebsite.mindspark.in\/wordpress\/math-concepts\/#numbers\" class=\"otherc\">Numbers<\/a><\/div>\n<\/div>\n<p>[\/et_pb_text][et_pb_text admin_label=&#8221;Related Concepts<br \/>\n&#8221; _builder_version=&#8221;4.9.11&#8243; _module_preset=&#8221;default&#8221; header_font=&#8221;|700|||||||&#8221; header_font_size=&#8221;25px&#8221; text_orientation=&#8221;center&#8221; custom_margin=&#8221;0px||0px||true|false&#8221; custom_padding=&#8221;8px|15px|0px|15px|false|true&#8221; locked=&#8221;off&#8221; global_colors_info=&#8221;{}&#8221;]<\/p>\n<h1>Related Concepts<\/h1>\n<p>[\/et_pb_text][et_pb_text _builder_version=&#8221;4.13.1&#8243; _module_preset=&#8221;default&#8221; text_line_height=&#8221;2.2em&#8221; link_font_size=&#8221;16px&#8221; custom_margin=&#8221;||0px||false|false&#8221; custom_padding=&#8221;10px|15px|10px|28px|true|false&#8221; hover_enabled=&#8221;0&#8243; locked=&#8221;off&#8221; global_colors_info=&#8221;{}&#8221; sticky_enabled=&#8221;0&#8243;]<\/p>\n<div>\n<div class=\"trr\"><a href=\"https:\/\/mindspark.in\/studymaterial\/math-concepts\/arithmetic-progression\/\" class=\"otherc\">Arithmetic Progression<\/a><\/div>\n<div class=\"trr\"><a href=\"https:\/\/mindspark.in\/studymaterial\/math-concepts\/arithmetic-progression-and-geometric-progression\/\" class=\"otherc\">AP and GP<\/a><\/div>\n<div class=\"trr\"><a href=\"https:\/\/mindspark.in\/studymaterial\/math-concepts\/sum-of-an-infinite-arithmetic-progression-mindspark\/\" class=\"otherc\">Sum of an infinite AP<\/a><\/div>\n<\/div>\n<p>[\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=&#8221;Row for space&#8221; _builder_version=&#8221;4.10.6&#8243; _module_preset=&#8221;default&#8221; locked=&#8221;off&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_column type=&#8221;4_4&#8243; _builder_version=&#8221;4.9.11&#8243; _module_preset=&#8221;default&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_divider show_divider=&#8221;off&#8221; _builder_version=&#8221;4.10.4&#8243; _module_preset=&#8221;default&#8221; 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_builder_version=&#8221;4.9.11&#8243; _module_preset=&#8221;default&#8221; global_colors_info=&#8221;{}&#8221;][et_pb_text admin_label=&#8221;FAQ&#8221; module_class=&#8221;faqstyl&#8221; _builder_version=&#8221;4.11.3&#8243; _module_preset=&#8221;default&#8221; text_font_size=&#8221;16px&#8221; header_font=&#8221;|700|||||||&#8221; header_text_align=&#8221;center&#8221; header_line_height=&#8221;2.5em&#8221; background_color=&#8221;#dbedc6&#8243; max_width=&#8221;80%&#8221; module_alignment=&#8221;center&#8221; custom_margin=&#8221;||||false|false&#8221; custom_padding=&#8221;30px|25px|30px|25px|true|true&#8221; border_radii=&#8221;on|10px|10px|10px|10px&#8221; global_colors_info=&#8221;{}&#8221;]<\/p>\n<h1>Frequently Asked Questions<span style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">\u00a0<\/span><\/span><\/h1>\n<ol>\n<li>What is an arithmetic progression?<\/li>\n<\/ol>\n<p>A series in which the difference between any two adjacent terms is constant is known as an arithmetic progression.<\/p>\n<ol start=\"2\">\n<li>What is the sum of first n numbers in an AP?<\/li>\n<\/ol>\n<p>The sum of first n numbers can be calculated using the formula S= (n[2a+(n-1)d])\/2<\/p>\n<ol start=\"3\">\n<li>What is the formula for the nth term of an AP?<\/li>\n<\/ol>\n<p>The formula to calculate the n<sup style=\"font-size: 10px;\">th<\/sup> term of an AP is T<sub style=\"font-size: 10px;\">n<\/sub>= a+(n-1)d<\/p>\n<ol start=\"4\">\n<li>How to calculate the number of terms in an Arithmetic Progression?<\/li>\n<\/ol>\n<p>The formula to calculate the number of terms in an AP is n= ((l-a))\/d+1[\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The nth term of an AP can be calculated using the formula  Tn = a+(n-1)d and the nth term from the end of the AP can be calculated using Tn = l-(n-1)d<\/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":"<!-- This site is optimized with the Yoast SEO plugin v17.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>nth term of an Arithmetic Progression - mydomain<\/title>\n<meta name=\"description\" content=\"The nth term of an AP can be calculated using the formula Tn= a+(n-1)d and the nth term from the end of the AP can be calculated using Tn= l-(n-1)d\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/nth-term-of-an-arithmetic-progression\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" 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