Question: What is the sum of the first 20 term of the sequence 0.7, 0.77, 0.777, … ?

A. 7/81(179−10⁽⁻²⁰⁾)
B. 7/9(99−10⁽⁻²⁰⁾)
C. 7/81(179+10⁽⁻²⁰⁾)
D. 7/9(99+10⁽⁻²⁰⁾)
E. 7/9(99+10⁽⁻²⁰⁾)

Answer: C

Solution and Explanation:

Approach Solution 1:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
0.7+0.77+0.777..…
= 7[(1/10)+(11/100)+(111/1000).......20 terms]
= 7/9[(9/10)+(99/100)+(999/1000)....20 terms]
= 7/9[(1-1/10)+(1-1/100)+(1-1/1000)......20 terms]
= 7/9[20- {(1/10)+(1/100)+(1/1000)+.....20 terms]
=7/9[20- {(1/10)(1-(1/10)²⁰)/9/10}
= 7/9[20- (1/9) +(1/9)*10^-20]
= 7/9[(179/9) + (1/9)*10^-20]
= 7/81[179+10^-20]
Correct option: C

Approach Solution 2:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Remember that 7/9 equals 0.777, 7/90 equals 0.0777, 7/900 is 0.00777, and so forth. We thus have:
The first term is 0.7 = 0.777... - 0.0777... = 7/9 - 7/90 = 7/9 - 7/9 x 10-1
Second term equals 0.77 = 0.777... - 0.00777... = 7/9 - 7/900 = 7/9 - 7/9 x 10-2
Third term: 0.777 = 0.777...-0.000777... = 7/9 - 7/9000 = 7/9 - 7/9 x 10-3
7/9 - 7/9 x 10-20 is the 20th term.
The total of the first 20 words is thus:
=7/9 x 20 - 7/9 x (10^-1 + 10^-2 + 10^-3 + … + 10^-20)
= 7/9 x [20 - (10^-1 + 10^-2 + 10^-3 + … + 10^-20)]
= 7/9 x (20 - 0.11...11) (20 - 0.11...11) (Note: There are 20 ones after the decimal point.)
= 7/81 x 9(20 - 0.11...11) (20 - 0.11...11)
= 7/81 x (180 - 9(0.11...11))
= 7/81 x (180 - 0.99...99) (180 - 0.99...99) (Note: There are 20 nines after the decimal point.) 7/81 x (180 – 1 + 0.00...01) Between the decimal point and the rightmost one, there are 19 zeros.
= 7/81 x (179 + 10^-20)
Correct option: C

Approach Solution 3:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Let n = 0.7 + 0.77 + 0.777 + … + 0.77...77 (Note: There are 20 sevens after the decimal point in the last word.)
So, we are trying to find n's value. When we divide the equation by 10, we get:
10n = 7 + 7.7 + 7.77 + … + 7.77...77 (Note: There are 20 sevens after the decimal point in the last word.)
When we deduct the first equation from the second, we get:
9n = 7 + 7 + 7 + … + 7 - 0.77...77 (Note: The right hand side of the equation comprises 21 terms, of which the last one is the final term from the previous equation; the first 20 terms are all 7.)
This can be rewritten as:
9n = 140 - 7(0.11...11) (0.11...11) (Note: There are 20 ones after the decimal point.)
9n = 7(20 - 0.11...11) (20 - 0.11...11)
n = 7/9 x (20 - 0.11...11) (20 - 0.11...11) (Note: The third line of the lengthy process in the first solution is on the right-hand side. Thus, we will likewise receive the final result of 7/81 x (179 + 10-20).
Correct option: C

“What is the sum of the first 20 term of the sequence 0.7, 0.77, 0.777," - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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