Problem 79 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The largest term of the sequence \(\frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}, \ldots\) is (A) \(\frac{16}{692}\) (B) \(\frac{4}{524}\) (C) \(\frac{49}{1529}\) (D) None of these

Step-by-Step Solution

Answer

The largest term is \(\frac{49}{1529}\).

Step 1: Express the General Term

Notice that the numerators are consecutive perfect squares: \(1^2, 2^2, 3^2, 4^2, \ldots\). We can express the general term of the sequence as \(\frac{n^2}{a_n}\) where \(n = 1, 2, 3, \ldots\). Each \(a_n\) corresponds to the denominator of the \(n\)-th term in the sequence.

Step 2: Identify the Pattern in Denominators

Observe the pattern in the denominators: 503, 524, 581, 692, etc. The increment does not follow a simple arithmetic pattern, but we can infer that they are each distinctly specific to their term. The formula for exact denominators may not be discernible without more terms, so we proceed by calculating each term's value.

Step 3: Calculate Each Term's Value

Calculate the values of the given terms: - For \(\frac{1}{503}\), calculate \(\approx 0.002\).- For \(\frac{4}{524}\), calculate \(\approx 0.00763\).- For \(\frac{9}{581}\), calculate \(\approx 0.01548\).- For \(\frac{16}{692}\), calculate \(\approx 0.02312\).- For \(\frac{49}{1529}\), calculate \(\approx 0.03204\).

Step 4: Compare Values to Find the Largest Term

Compare the calculated term values: - \(0.002 < 0.00763 < 0.01548 < 0.02312 < 0.03204\). The largest value corresponds to the term \(\frac{49}{1529}\).

Key Concepts

Denominator Pattern

In the sequence provided, each term is written in the form \( \frac{n^2}{a_n} \). Here, \( n^2 \) signifies the numerator and \( a_n \) the denominator. Unlike the numerators, which are clearly perfect squares, the denominators follow a more elusive pattern. As you progress through the sequence, you naturally notice that these denominators do not increase by the same amount each time. This means there is no simple arithmetic progression we can detect immediately.

Yet, even though a precise formula for these denominators isn't readily apparent, it’s important to identify that these intriguing numbers are specifically tied to their respective terms in the sequence. Recognizing that each step includes a seemingly random yet deliberate increase is key. When analyzing such patterns, attempting to decipher them without a complete formula requires looking at other aspects of the sequence. In this problem, since the denominators may not provide enough clarity, focus shifts to calculating the value of each term to find the largest one.

Perfect Squares

Perfect squares are numbers like 1, 4, 9, 16, etc., that are the result of squaring integers. In this sequence, each numerator is a perfect square. For instance:

• The first term's numerator is \(1\), which is \(1^2\).
• The second term is \(4\), equal to \(2^2\).
• The third term has the numerator \(9\), or \(3^2\).
• Following this logic, the sequence extends with \(16\) and \(49\), represented by \(4^2\) and \(7^2\), respectively.

These perfect squares grow rapidly as you increase the sequence. They play a critical role in determining the size of each term when their denominators are taken into account. In sequences, identifying perfect squares quickly can help predict or verify the properties of terms and understand their comparative sizes more easily.

Largest Term in Sequence

The goal is to identify which term in the sequence has the highest value. To do this, we must calculate the actual numerical values of each term. By forming fractions with perfect squares in the numerators, we determine their relative sizes by comparing the results.- Begin by calculating each of the terms given in the problem:

• \( \frac{1}{503} \approx 0.002 \)
• \( \frac{4}{524} \approx 0.00763 \)
• \( \frac{9}{581} \approx 0.01548 \)
• \( \frac{16}{692} \approx 0.02312 \)
• \( \frac{49}{1529} \approx 0.03204 \)

- Once these values are computed, simply comparing them reveals that \( \frac{49}{1529} \) is the largest because it has the highest value of \( 0.03204 \).

This method highlights the importance of direct calculation in sequences with non-obvious patterns, as it allows you to clearly see the differences in magnitude between sequential terms.