The given series is:7.8\u203e+7.88\u203e+7.888\u203e+7.8888\u203e+\u22ef7.\\overline{8} + 7.\\overline{88} + 7.\\overline{888} + 7.\\overline{8888} + \\cdots7.8+7.88+7.888+7.8888+\u22ef<\/p>\n\n\n\n
Each term in the series is of the form 7.8\u2026\u203e7.\\overline{8 \\ldots}7.8\u2026 with an increasing number of 8’s in the repeating decimal.<\/p>\n\n\n\n
To solve this, we break it down into the form of a geometric series. The general term for this type of recurring decimal can be written as:Sn=7+89\u22c510\u2212nS_n = 7 + \\frac{8}{9} \\cdot 10^{-n}Sn\u200b=7+98\u200b\u22c510\u2212n<\/p>\n\n\n\n
Where:<\/p>\n\n\n\n
- \n
- SnS_nSn\u200b represents the nnn-th term in the series.<\/li>\n\n\n\n
- The denominator 9 is used because of the repeating decimal (as 0.\\overline{8} = 8\/9).<\/li>\n\n\n\n
- The term 10\u2212n10^{-n}10\u2212n is used to account for the position of the decimal point shifting with each subsequent term.<\/li>\n<\/ul>\n\n\n\n
Step-by-Step Calculation:<\/h3>\n\n\n\n
- \n
- First term (7.\\overline{8})<\/strong>: 7.8\u203e=7+897.\\overline{8} = 7 + \\frac{8}{9}7.8=7+98\u200b which is the sum of a constant and the fraction from the recurring decimal.<\/li>\n\n\n\n
- Second term (7.\\overline{88})<\/strong>: 7.88\u203e=7+88997.\\overline{88} = 7 + \\frac{88}{99}7.88=7+9988\u200b This term has two repeating digits, so the denominator becomes 99.<\/li>\n\n\n\n
- Third term (7.\\overline{888})<\/strong>: 7.888\u203e=7+8889997.\\overline{888} = 7 + \\frac{888}{999}7.888=7+999888\u200b And so on, as the number of repeating digits increases.<\/li>\n<\/ol>\n\n\n\n
The series can be expressed as the sum:\u2211n=12014(7+8\u22ef89\u22ef9)\\sum_{n=1}^{2014} \\left( 7 + \\frac{8 \\cdots 8}{9 \\cdots 9} \\right)n=1\u22112014\u200b(7+9\u22ef98\u22ef8\u200b)<\/p>\n\n\n\n
where the number of digits in the numerator and denominator increases as we progress.<\/p>\n\n\n\n
Summing the Infinite Series:<\/h4>\n\n\n\n
To calculate the value of this sum up to 2014 terms, we treat it as an infinite geometric series. The formula for the sum of an infinite geometric series is:S=a1\u2212rS = \\frac{a}{1 – r}S=1\u2212ra\u200b<\/p>\n\n\n\n
Where:<\/p>\n\n\n\n
- \n
- aaa is the first term,<\/li>\n\n\n\n
- rrr is the common ratio between successive terms.<\/li>\n<\/ul>\n\n\n\n
Since the series has an increasing number of 8’s in each term, it becomes more complicated. However, it can be approximated with the following formula:S\u22482014\u22c5(7+89) (approximate)S \\approx 2014 \\cdot \\left( 7 + \\frac{8}{9} \\right) \\text{ (approximate)}S\u22482014\u22c5(7+98\u200b) (approximate)<\/p>\n\n\n\n
This simplifies to:S\u22482014\u22c5(7.8888)S \\approx 2014 \\cdot \\left( 7.8888 \\right)S\u22482014\u22c5(7.8888)<\/p>\n\n\n\n
Finally, the total sum up to 2014 terms is:S\u224815859.51S \\approx 15859.51S\u224815859.51<\/p>\n\n\n\n
This is the approximate value of the sum of the given series up to 2014 terms.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-228.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"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 series is:7.8\u203e+7.88\u203e+7.888\u203e+7.8888\u203e+\u22ef7.\\overline{8} + 7.\\overline{88} + 7.\\overline{888} + 7.\\overline{8888} + \\cdots7.8+7.88+7.888+7.8888+\u22ef Each term in the series is of the form 7.8\u2026\u203e7.\\overline{8 \\ldots}7.8\u2026 […]<\/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-248424","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248424","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=248424"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248424\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248424"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248424"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Second term (7.\\overline{88})<\/strong>: 7.88\u203e=7+88997.\\overline{88} = 7 + \\frac{88}{99}7.88=7+9988\u200b This term has two repeating digits, so the denominator becomes 99.<\/li>\n\n\n\n
- First term (7.\\overline{8})<\/strong>: 7.8\u203e=7+897.\\overline{8} = 7 + \\frac{8}{9}7.8=7+98\u200b which is the sum of a constant and the fraction from the recurring decimal.<\/li>\n\n\n\n