{ "@context": "https://schema.org/", "@type": "Quiz", "about": { "@type": "Thing", "name": "Geometrical Progression" }, "educationalAlignment": [ { "@type": "AlignmentObject", "alignmentType": "educationalSubject", "targetName": "Mathematics" } ], "hasPart": [ { "@context": "https://schema.org/", "@type": "Question", "eduQuestionType": "Flashcard", "text": "Evaluate (0.7+0.77+0.777+...+upto n times)", "acceptedAnswer": { "@type": "Answer", "text": "{7}/{9}(0.9 + 0.99 + 0.999 + ... + upto n times) = {7}/{9}({9}/{10} + {99}/{100} + {999}/{1000} + ... + upto n times)= {7}/{9}(1-{1}/{10} + 1-{1}/{100} + 1-{1}/{1000} + ... + upto n times)= {7}/{9}(1-{1}/{10^1} + 1-{1}/{10^2} + 1-{1}/{10^3} + ... +upto n times)= {7}/{9}[n -{1}/{10^1} + {1}/{10^2} + {1}/{10^3} + ... + upto n times)] {7}/{9}[n - {1}/{10} * ({1}/{10})^n - 1)}/{{1}/{10} - 1}]= {7}/{9}[n - {{1}/{10} * (({1}/{10^n}) - 1)}{{-9}/{10}}]= {7}/{9}[n +({1}/{9} * ({1}/{10^n} - 1))" } } ] } { "@context": "http://schema.org", "@type": "QAPage", "name": "Evaluate (0.7+0.77+0.777+...+upto n times)", "description": "= {7}{9}(0.9 + 0.99 + 0.999 + ... + upto n times) = {7}{9}({9}{10} + {99}{100} + {999}{1000} + ... + upto n times)= {7}{9}(1-{1}{10} + 1-{1}{100} + 1-{1}{1000} + ... + upto n times)= {7}{9}(1-{1}{10^1} + 1-{1}{100^2} + 1-{1}{1000^3} + ... +upto n times)= {7}{9}[n -{1}{10^1} + {1}{100^2} + {1}{1000^3} + ... + upto n times)]...", "mainEntity": { "@type": "Question", "@id": "https://readaxis.com/question-answer/if-xyz12-x2y2z296-and-1-x1-y1-z36-find-the-value-of-x3y3z3-4/", "name": "Evaluate (0.7+0.77+0.777+...+upto n times)", "text": "Evaluate (0.7+0.77+0.777+...+upto n times)", "dateCreated": "2023-01-19T11:52Z", "answerCount": "1", "author": { "@type": "Person", "name": "ReadAxis", "url": "https://www.readaxis.com/" }, "acceptedAnswer": { "@type": "Answer", "upvoteCount": "2", "text": "{7}/{9}(0.9 + 0.99 + 0.999 + ... + upto n times) = {7}/{9}({9}/{10} + {99}/{100} + {999}/{1000} + ... + upto n times)= {7}/{9}(1-{1}/{10} + 1-{1}/{100} + 1-{1}/{1000} + ... + upto n times)= {7}/{9}(1-{1}/{10^1} + 1-{1}/{10^2} + 1-{1}/{10^3} + ... +upto n times)= {7}/{9}[n -{1}/{10^1} + {1}/{10^2} + {1}/{10^3} + ... + upto n times)] {7}/{9}[n - {1}/{10} * ({1}/{10})^n - 1)}/{{1}/{10} - 1}]= {7}/{9}[n - {{1}/{10} * (({1}/{10^n}) - 1)}{{-9}/{10}}]= {7}/{9}[n +({1}/{9} * ({1}/{10^n} - 1))", "url": "https://readaxis.com/question-answer/if-xyz12-x2y2z296-and-1-x1-y1-z36-find-the-value-of-x3y3z3-4/", "dateCreated": "2023-01-19T11:52Z", "author": { "@type": "Person", "name": "Readaxis Admin" } }, "suggestedAnswer": [] } }

Question

Evaluate $$(0.7 + 0.77 + 0.777 + … + \:upto\; n \;times)$$

Solution:

$$7 (0.1 + 0.11 + 0.111 + …. +$$upto n times)
$$= \frac{7}{9}(0.9 + 0.99 + 0.999 + … +$$upto n times)
$$= \frac{7}{9}(\frac{9}{10} + \frac{99}{100} + \frac{999}{1000} + … +$$ upto n times)
$$= \frac{7}{9}(1-\frac{1}{10} + 1-\frac{1}{100} + 1-\frac{1}{1000} + … +$$ upto n times)
$$= \frac{7}{9}(1-\frac{1}{10^1} + 1-\frac{1}{10^2} + 1-\frac{1}{10^3} + … +$$upto n times)
$$= \frac{7}{9}[n -(\frac{1}{10^1} + \frac{1}{10^2} + \frac{1}{10^3} + … +$$upto n times)]

By Sum of GP having $$a= \frac{1}{10}\: and\: r = \frac{1}{10}$$

$$= \frac{7}{9}[n\; – \frac{\frac{1}{10} \times ((\frac{1}{10})^n – 1)}{\frac{1}{10} – 1}]$$
$$= \frac{7}{9}[n \;- \frac{\frac{1}{10} \times ((\frac{1}{10^n}) – 1)}{\frac{-9}{10}}]$$
$$= \frac{7}{9}[n \;+(\frac{1}{9} \times (\frac{1}{10^n} – 1))$$