{"id":248392,"date":"2025-07-08T10:48:29","date_gmt":"2025-07-08T10:48:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248392"},"modified":"2025-07-08T10:48:31","modified_gmt":"2025-07-08T10:48:31","slug":"find-the-value-of-7-8-bar-7-88-bar-7-888-bar-7-8888-bar-up-to-2014-times","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/find-the-value-of-7-8-bar-7-88-bar-7-888-bar-7-8888-bar-up-to-2014-times\/","title":{"rendered":"find the value of 7.8 bar +7.88 bar +7.888 bar + 7.8888 bar + \u2026.. up to 2014 times"},"content":{"rendered":"\n<p>find the value of 7.8 bar +7.88 bar +7.888 bar + 7.8888 bar + \u2026.. up to 2014 times (interns of non terminating recurring decimal form<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The given problem involves summing a sequence of numbers, each with a recurring decimal. The sequence is:7.8\u203e+7.888\u203e+7.8888\u203e+7.88888\u203e+\u20267.\\overline{8} + 7.88\\overline{8} + 7.888\\overline{8} + 7.8888\\overline{8} + \\dots7.8+7.888+7.8888+7.88888+\u2026<\/p>\n\n\n\n<p>The sequence continues with each successive term having one more &#8220;8&#8221; in the decimal places and the number of terms is 2014.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Converting each term into a non-terminating recurring decimal<\/h3>\n\n\n\n<p>Let\u2019s express each term in the sequence as a non-terminating recurring decimal:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>7.8\u203e=7+0.8\u203e7.\\overline{8} = 7 + 0.\\overline{8}7.8=7+0.8, which is a repeating decimal.<\/li>\n\n\n\n<li>7.888\u203e=7+0.888\u203e7.88\\overline{8} = 7 + 0.88\\overline{8}7.888=7+0.888.<\/li>\n\n\n\n<li>7.8888\u203e=7+0.8888\u203e7.888\\overline{8} = 7 + 0.888\\overline{8}7.8888=7+0.8888, and so on.<\/li>\n<\/ul>\n\n\n\n<p>Each term can be written in the form:7+89,\u20097+890,\u20097+8900,\u20267 + \\frac{8}{9}, \\, 7 + \\frac{8}{90}, \\, 7 + \\frac{8}{900}, \\dots7+98\u200b,7+908\u200b,7+9008\u200b,\u2026<\/p>\n\n\n\n<p>The repeating part 8\u203e\\overline{8}8 in decimal corresponds to the fraction 89\\frac{8}{9}98\u200b, but the denominator changes depending on how many digits are repeated after the decimal point.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Recognizing the general formula for each term<\/h3>\n\n\n\n<p>The general form for the nnn-th term in the sequence is:7+810n\u221217 + \\frac{8}{10^n &#8211; 1}7+10n\u221218\u200b<\/p>\n\n\n\n<p>Where nnn is the number of digits after the decimal. For the first term, n=1n = 1n=1, for the second term n=2n = 2n=2, and so on up to n=2014n = 2014n=2014.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Summing the sequence<\/h3>\n\n\n\n<p>The sum can be written as:S=\u2211n=12014(7+810n\u22121)S = \\sum_{n=1}^{2014} \\left( 7 + \\frac{8}{10^n &#8211; 1} \\right)S=n=1\u22112014\u200b(7+10n\u221218\u200b)<\/p>\n\n\n\n<p>This sum can be split into two parts:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The constant part: \u2211n=120147=7\u00d72014=14098\\sum_{n=1}^{2014} 7 = 7 \\times 2014 = 14098\u2211n=12014\u200b7=7\u00d72014=14098<\/li>\n\n\n\n<li>The fractional part: \u2211n=12014810n\u22121\\sum_{n=1}^{2014} \\frac{8}{10^n &#8211; 1}\u2211n=12014\u200b10n\u221218\u200b<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Evaluating the fractional sum<\/h3>\n\n\n\n<p>The sum of the fractional parts involves a series of terms where each term decreases as nnn increases. This series converges rapidly, and we can approximate the sum of the fractions. For large nnn, the term 810n\u22121\\frac{8}{10^n &#8211; 1}10n\u221218\u200b becomes very small.<\/p>\n\n\n\n<p>Thus, the total sum is:S\u224814098+\u2211n=12014810n\u22121S \\approx 14098 + \\sum_{n=1}^{2014} \\frac{8}{10^n &#8211; 1}S\u224814098+n=1\u22112014\u200b10n\u221218\u200b<\/p>\n\n\n\n<p>The sum of the fractions \u2211n=12014810n\u22121\\sum_{n=1}^{2014} \\frac{8}{10^n &#8211; 1}\u2211n=12014\u200b10n\u221218\u200b is roughly equal to:\u2211n=12014810n\u22480.8888\\sum_{n=1}^{2014} \\frac{8}{10^n} \\approx 0.8888n=1\u22112014\u200b10n8\u200b\u22480.8888<\/p>\n\n\n\n<p>So, the total sum SSS is approximately:S\u224814098+0.8888=14098.8888S \\approx 14098 + 0.8888 = 14098.8888S\u224814098+0.8888=14098.8888<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>S\u224814098.8888S \\approx 14098.8888S\u224814098.8888<\/p>\n\n\n\n<p>Thus, the value of the series up to 2014 terms is approximately <strong>14098.8888<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-220.jpeg\" alt=\"\" class=\"wp-image-248393\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>find the value of 7.8 bar +7.88 bar +7.888 bar + 7.8888 bar + \u2026.. up to 2014 times (interns of non terminating recurring decimal form The Correct Answer and Explanation is: The given problem involves summing a sequence of numbers, each with a recurring decimal. The sequence is:7.8\u203e+7.888\u203e+7.8888\u203e+7.88888\u203e+\u20267.\\overline{8} + 7.88\\overline{8} + 7.888\\overline{8} + 7.8888\\overline{8} [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-248392","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=248392"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248392\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}