Problem 157 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The sum of first 20 terms of the sequence \(0.7,0.77\), \(0.777, \ldots\) is (A) \(\frac{7}{9}\left(99-10^{-20}\right)\) (B) \(\frac{7}{81}\left(179+10^{-20}\right)\) (C) \(\frac{7}{9}\left(99+10^{-20}\right)\) (D) \(\frac{7}{81}\left(179-10^{-20}\right)\)

Step-by-Step Solution

Answer

Option (D) is correct: \(\frac{7}{81}(179 - 10^{-20})\).

Step 1: Identify the Sequence

The given sequence is \(0.7, 0.77, 0.777, \ldots\). This suggests each term has repeated digits of '7', increasing in number by one in each subsequent term.

Step 2: Express nth Term in General Form

Notice the pattern: the 1st term is \(0.7 = \frac{7}{10}\), the 2nd term is \(0.77 = \frac{77}{100}\), and the 3rd term is \(0.777 = \frac{777}{1000}\). The nth term can be expressed as \(0.777\ldots7\) (n times) = \(\frac{7(10^n - 1)}{9 \times 10^n}\).

Step 3: Setup the Summation of the Series

We want to find the sum of the first 20 terms: \(S_n = \sum_{n=1}^{20} \frac{7(10^n - 1)}{9 \times 10^n}\). This simplifies to \(S_n = \frac{7}{9} \sum_{n=1}^{20} \left(1 - \frac{1}{10^n}\right)\).

Step 4: Simplify the Summation Formula

The sum can be broken into two parts: \(\sum_{n=1}^{20} 1 = 20\) and \(\sum_{n=1}^{20} \frac{1}{10^n}\) which is a geometric series with first term \(\frac{1}{10}\) and common ratio \(\frac{1}{10}\), giving: \(\frac{1}{10} \cdot \frac{1-(\frac{1}{10})^{20}}{1 - \frac{1}{10}}\).

Step 5: Evaluate the Components

The sum of the constant 1 part is 20. The sum of the geometric series is \(\frac{1}{9} (1 - 10^{-20})\).

Step 6: Compute the Total Sum

Substituting back, we get \(S_n = \frac{7}{9} \left(20 - \frac{1}{9}(1 - 10^{-20})\right)\). Simplifying these gives \(\frac{7}{9} \left(\frac{179}{9} - 10^{-20}\right)\).

Step 7: Final Simplified Result

The simplified expression is \(\frac{7}{81} (179 - 10^{-20})\). Thus, the answer is option (D).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Understanding this kind of series is crucial because they appear in many mathematical situations. When you have a geometric series, it follows a pattern that makes it easier to find sums and other properties.
In the given exercise, the portion dealing with \( \sum_{n=1}^{20} \frac{1}{10^n} \), is an example of a geometric series. Here, the first term is \( \frac{1}{10} \), and the common ratio is also \( \frac{1}{10} \). Therefore, the sum of this series can be calculated using the formula for the sum of a geometric series:

• First term, \( a = \frac{1}{10} \).
• Common ratio, \( r = \frac{1}{10} \).
• Number of terms, \( n = 20 \).

Using the formula \[ S_n = a \frac{1-r^n}{1-r} \], we compute the sum of this geometric series. It's straightforward when you know each part of the series and how they interact.

Summation Formula

For many sequences and series, the use of summation formulas can simplify complex calculations and reduce errors during computation.
A summation formula allows us to easily calculate the total of a series without having to manually add up each term. In this example, the problem uses a specific summation formula to find the sum of the first 20 terms: \( S_n = \frac{7}{9} \sum_{n=1}^{20} \left(1 - \frac{1}{10^n}\right) \).
Let's break it down a bit:

• The term \( 1 - \frac{1}{10^n} \) is a simplification that accounts for both the constant part and the geometric series part of the summation.
• The factor \( \frac{7}{9} \) is constant across all terms, indicating that every term of the series is multiplied by this factor.

The expression \( \sum_{n=1}^{20} 1 \) gives us the constant sum of 20, while \( \sum_{n=1}^{20} \frac{1}{10^n} \) is calculated using the geometric series formula. Both of these are combined to find the total sum in a more efficient manner.

Sequence Pattern

Recognizing a sequence pattern is important in identifying the relationship between terms and deriving formulas for them. A sequence pattern essentially describes how each subsequent term relates to the previous one based on a simple rule or operation.
In the provided exercise, the sequence was identified from the pattern: \( 0.7, 0.77, 0.777, \ldots \). This indicates that each number is made by repeating the digit '7' an increasing number of times. This pattern can be described in the general form:

• The 1st term is \( 0.7 = \frac{7}{10} \).
• The 2nd term is \( 0.77 = \frac{77}{100} \).
• The nth term is \( \text{0.777}\ldots\text{7 (n times)} = \frac{7(10^n - 1)}{9 \times 10^n} \).

This pattern recognition takes advantage of a mathematical concept where repeating decimals can be expressed as fractions. Identifying and expressing this pattern accurately allows you to set up and solve series-related problems more effectively.