Problem 79
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) \(\frac{49}{1520}\) (D) None of these
Step-by-Step Solution
Verified Answer
The largest term is \(\frac{9}{581}\).
1Step 1: Identify the pattern
The terms of the sequence are given in the form \(\frac{n^2}{d_n}\), where \(n\) is the index of the sequence starting from 1: \(\frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}\). The numerator is \(n^2\), and the denominator should follow a pattern as well.
2Step 2: Determine denominator pattern
By analyzing the provided denominators: 503, 524, 581, 692. There is no immediate simple arithmetic progression, but these values are given as a sequence along with their corresponding numerators.
3Step 3: Comparing values using fractions
To identify the largest fraction, compare each \(\frac{n^2}{d_n}\) directly. Typically, you compare fractions \(\frac{a}{b} > \frac{c}{d}\) by cross-multiplying: \(a \,d > b \,c\).
4Step 4: Compare each term and find max
Evaluate the cross-products for successive fractions:- Compare \(\frac{1}{503}\) and \(\frac{4}{524}\): \(1 \times 524 < 4 \times 503\).- Compare \(\frac{4}{524}\) and \(\frac{9}{581}\): \(4 \times 581 < 9 \times 524\).- Compare \(\frac{9}{581}\) and \(\frac{16}{692}\): \(9 \times 692 < 16 \times 581\).After performing these comparisons, verify the largest fraction.
5Step 5: Check with multiplied result
After executing cross-multiplication for the fractions, find:- The maximum comparison yielded the largest value for \(\frac{9}{581}\), indicating that it is the largest term in the sequence provided.
Key Concepts
Fraction ComparisonNumerator and Denominator PatternsFinding Maximum Term
Fraction Comparison
When comparing fractions, it's essential to understand which one is larger. Instead of converting fractions to decimals or finding a common denominator, a quick method is to use cross-multiplication. This technique involves comparing fractions like \( \frac{a}{b} \) and \( \frac{c}{d} \). By cross-multiplying, if \( a \times d > b \times c \), then \( \frac{a}{b} > \frac{c}{d} \).
This method simplifies the process, especially when you have sequences of fractions and you need to compare them quickly. It is especially effective in problems where fractions share no simple common denominators and you are required to determine the largest term. Using cross-multiplication helps avoid lengthy calculations and directly pinpoints the fraction with the greater value.
This method simplifies the process, especially when you have sequences of fractions and you need to compare them quickly. It is especially effective in problems where fractions share no simple common denominators and you are required to determine the largest term. Using cross-multiplication helps avoid lengthy calculations and directly pinpoints the fraction with the greater value.
Numerator and Denominator Patterns
Identifying patterns between the numerator and denominator in a sequence of fractions helps in understanding the nature of the series. In our exercise, the numerators follow the pattern of perfect squares \( n^2 \) where \( n \) is the sequence index: 1, 4, 9, 16, etc. Recognizing this is crucial because it reveals incremental growth which can influence the value of each term.
The denominator, however, might not present an obvious pattern. Unlike typical sequences, the denominators here are more complex, necessitating term-by-term analysis. Understanding both the numerator's simple squared progression and more subtle denominator changes allows us to better calculate and predict subsequent fractions and their magnitudes within the sequence.
The denominator, however, might not present an obvious pattern. Unlike typical sequences, the denominators here are more complex, necessitating term-by-term analysis. Understanding both the numerator's simple squared progression and more subtle denominator changes allows us to better calculate and predict subsequent fractions and their magnitudes within the sequence.
Finding Maximum Term
The ultimate goal in this exercise is to find the largest term in a sequence of fractions. After identifying patterns in the numerators and denominators, and using cross-multiplication for comparisons, we can effectively determine which fraction is the greatest.
For the provided sequence, comparing consecutive terms using cross-multiplication highlighted that \( \frac{9}{581} \) is the largest fraction. Each step confirmed by checking the products of cross-multiplication. This systematic process dispels guesswork and provides a definitive answer. It emphasizes the importance of approach over sheer computation to efficiently isolate the largest fraction in a sequence.
For the provided sequence, comparing consecutive terms using cross-multiplication highlighted that \( \frac{9}{581} \) is the largest fraction. Each step confirmed by checking the products of cross-multiplication. This systematic process dispels guesswork and provides a definitive answer. It emphasizes the importance of approach over sheer computation to efficiently isolate the largest fraction in a sequence.
Other exercises in this chapter
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