Problem 128 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The sum of first 20 terms of the sequence \(0.7,0.77,0.777, \ldots .\) is \(\quad[2013]\) (a) \(\frac{7}{81}\left(179-10^{-20}\right)\) (b) \(\frac{7}{9}\left(99-10^{-20}\right)\) (c) \(\frac{7}{81}\left(179+10^{-20}\right)\) (d) \(\frac{7}{9}\left(99+10^{-20}\right)\)

Step-by-Step Solution

Answer

(c) \(\frac{7}{81}\left(179+10^{-20}\right)\)

Step 1: Identify the Sequence Pattern

Notice that the sequence is given as \(0.7, 0.77, 0.777, \ldots\). Each term in the sequence can be viewed as a repeated decimal. For the \(n\)-th term, the decimal appears \(n\) times.

Step 2: Express the General Term

Each term in the sequence can be expressed as a geometric series. For instance, the first term 0.7 is \(\frac{7}{10}\), the second term 0.77 is \(\frac{77}{100}\), and so on. Generally, the \(n\)-th term is \(\frac{7}{9}(1 - 10^{-n})\).

Step 3: Write the Sum of First 20 Terms

To find the sum of the first 20 terms, we sum the general term: \[ S_{20} = \sum_{n=1}^{20} \frac{7}{9}(1 - 10^{-n}) \] This simplifies to \[ S_{20} = \frac{7}{9}\left(\sum_{n=1}^{20}1 - \sum_{n=1}^{20}10^{-n}\right). \]

Step 4: Simplify the Sum

The first part \(\sum_{n=1}^{20}1\) is simply \(20\). The second part \(\sum_{n=1}^{20}10^{-n}\) is a geometric series: \[ \sum_{n=1}^{20}10^{-n} = \frac{0.1(1 - 10^{-20})}{1 - 0.1} = \frac{1}{9}(1 - 10^{-20}). \] Combining these results in: \[ S_{20} = \frac{7}{9}(20 - \frac{1}{9}(1 - 10^{-20})). \]

Step 5: Final Calculation

Substitute and calculate to simplify further: \[ S_{20} = \frac{7}{9}\left(20 - \frac{1}{9} + \frac{10^{-20}}{9}\right) = \frac{7}{9}\left(\frac{179}{9} + \frac{10^{-20}}{9}\right). \] Finally, \[ S_{20} = \frac{7}{81}\left(179 + 10^{-20}\right), \] matching option (c).

Key Concepts

geometric series

A geometric series is a sequence where each term is a constant multiple of the previous one. In the sequence provided, each term like 0.7, 0.77, and 0.777 can be expressed through geometric means. Let's explore this:

• First term, 0.7, is \( \frac{7}{10} \).
• Second term, 0.77, is \( \frac{77}{100} \) or \( \frac{7}{10} + \frac{7}{100} \).
• Third term, 0.777, is \( \frac{7}{10} + \frac{7}{100} + \frac{7}{1000}\).

Each term gets longer, with each new digit repeating the same pattern. Furthermore, we notice that each term sums smaller and smaller decimal values, forming a geometric series where the pattern is a repeated decimal extension.

sequence pattern

Identifying sequence patterns is crucial for understanding how terms are created and predicting future ones. In this problem, the sequence is 0.7, 0.77, 0.777, etc. We can see the following sequence pattern:

• The pattern involves adding one more 7 decimal each time.
• Expressing each number as a fraction in powers of 10 helps.

For the nth term, it evolves by having n occurrences of the digit 7 in decimal form, which gives us a general term derived from the sum of digit fractions. Recognizing this creates a clear picture of how each term grows within the series.

decimal representation

Decimal representation is a way to express numbers as sums of fractions, where the denominators are powers of 10. In the series 0.7, 0.77, 0.777, decimal representation helps simplify and understand:

• 0.7 becomes \( \frac{7}{10} \)
• 0.77 is simplified to \( \frac{7}{10} + \frac{7}{100} \)
• 0.777 translates to \( \frac{7}{10} + \frac{7}{100} + \frac{7}{1000} \)

By representing each term as a sum of fractions, it is easier to transform the recurring decimals into fraction form. This also leads us directly into recognizing geometric series, where each additional decimal represents a continuously decreasing fraction, much easier to manage in calculations.

summation formula

The summation formula helps calculate the sum of a sequence by laying down a mathematical process to follow. In this exercise, the series sums the first 20 terms:The formula used is:\[ S_{20} = \sum_{n=1}^{20} \frac{7}{9}(1 - 10^{-n}) \]This breaks into two parts:

• Sum of the whole parts, concise with 20.
• Sum of the remaining geometric series part, for fractional amounts.

The calculation then becomes simplified eloquently, using the geometric series formula:\[ \sum_{n=1}^{20} 10^{-n} = \frac{0.1(1 - 10^{-20})}{1 - 0.1} \]Each step loops back into a simple summation structure that adds clarity to the results computed. By following a summation formula, complex sequence sums reveal straightforward paths to getting results.