Problem 132 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The sum of the series : \(\quad\) [Online April 9, 2013] \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots \ldots\) upto 10 terms, is : (a) \(\frac{18}{11}\) (b) \(\frac{22}{13}\) (c) \(\frac{20}{11}\) (d) \(\frac{16}{9}\)

Step-by-Step Solution

Answer

The sum of the series up to 10 terms is \( \frac{16}{9} \) (Option d).

Step 1: Understand the series pattern

Observe the series: \( 1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots \). Each term begins with 1 in the numerator and a set of consecutive integers in the denominator. Specifically, the denominator of each term is the sum of natural numbers up to \( n \). For instance, the first term has just 1, the second term has \( 1+2 \), and so on.

Step 2: Identify the denominator pattern

The pattern in the denominator for each term is the sum of the first \( n \) natural numbers: \( 1, 1+2, 1+2+3, \ldots \). The sum of the first \( n \) natural numbers can be calculated using the formula: \( \frac{n(n+1)}{2} \).

Step 3: Write the first 10 terms

Calculate each term using the sum formula: \(1, \frac{1}{3}, \frac{1}{6}, \frac{1}{10}, \frac{1}{15}, \frac{1}{21}, \frac{1}{28}, \frac{1}{36}, \frac{1}{45}, \frac{1}{55} \).

Step 4: Calculate the sum of the first 10 terms

Add the first 10 terms: \[ 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \frac{1}{15} + \frac{1}{21} + \frac{1}{28} + \frac{1}{36} + \frac{1}{45} + \frac{1}{55} \]. Perform these calculations to get the sum.

Step 5: Simplify the sum

Perform successive addition of the fractions, finding a common denominator where needed. Simplify as much as possible to match one of the choices provided in the exercise.

Key Concepts

Sum of Natural Numbers

The sum of natural numbers is an essential concept in understanding sequences and series, and it plays a critical role in this exercise. Natural numbers are the basic building blocks of mathematics, and include numbers like 1, 2, 3, and so on, without end. When dealing with sequences, the sum of these numbers is often necessary to calculate overall totals or to identify patterns.

The formula for the sum of the first \(n\) natural numbers is \( \frac{n(n+1)}{2} \). This formula makes it easier to quickly find the total of natural numbers up to any given number, without having to add each one separately. For instance, to find the sum of the first three natural numbers, which are 1, 2, and 3, use the formula: \( \frac{3(3+1)}{2} = \frac{12}{2} = 6 \).

Understanding this formula is crucial not only for this exercise but also for tackling similar problems involving arithmetic series.

Fractions

Fractions are a way of expressing numbers that aren't whole. In this exercise, fractions are heavily utilized, especially in the form of \( \frac{1}{n(n+1)/2} \), reflecting parts of a whole for each term of the series.

Each fraction represents a part of the series you'll be summing up. Fractions consist of a numerator and a denominator. In this example, the numerator is always 1, which simplifies calculations considerably. The denominator, however, is based on the sum of natural numbers, making each term in the series distinct and influenced by the natural numbers' sum formula.

Adding fractions involves using a common denominator, which can require some extra calculations. By finding equivalent fractions for each term, where all the terms share the same denominator, you make it possible to add them together seamlessly. Mastering this is essential for working through problems like this that involve sequential addition of fractional terms.

Mathematical Patterns

Recognizing mathematical patterns is an important skill in mathematics, particularly when dealing with sequences and series like the one in this exercise. Patterns form the basis of predicting what will come next, simplifying the process of calculation.

In the given series, the pattern lies in the denominator of each term. Notice how each denominator is constructed by summing natural numbers consecutively: first 1, then 1+2, then 1+2+3, and so on. This clear pattern allows you to use the sum of natural numbers formula to calculate each term efficiently, without manually adding each set of numbers.

Understanding these patterns can simplify complex calculations, turning a long and potentially complicated process into a straightforward application of a formula.

Sequence and Series

Sequences and series are mathematical concepts that involve ordered lists of numbers and their summation, respectively. A sequence is simply a list of numbers arranged according to a specific rule. When these numbers are summed up, they form a series.

In this exercise, we deal with a specific type of series: an arithmetic series, where the sequence follows an identifiable pattern in the denominators. By identifying the rule governing the sequence—the sum of natural numbers—we can easily calculate each term in the series.

Working with sequences and series requires recognition of the underlying pattern and the ability to apply it across multiple terms. This process involves adding successive terms, in this case, fractions, to find the total sum. Understanding these concepts deeply will improve problem-solving efficiency and effectiveness in mathematics.