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Magazine Chapter 2: Problem 5
\(5-8\) Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. \(a_{n}=\frac{n^{2}}{2 n+3 n^{2}}\)
Short Answer
Expert verified
The sequence converges to \( \frac{1}{3} \).
Step by step solution
01
Understand the Sequence Formula
The given sequence is \( a_n = \frac{n^2}{2n + 3n^2} \). This formula will be used to calculate the first ten terms of the sequence by substituting values of \( n \) from 1 to 10.
02
Calculate the First Term
Substitute \( n = 1 \) into the sequence formula:\[a_1 = \frac{1^2}{2 \times 1 + 3 \times 1^2} = \frac{1}{2 + 3} = \frac{1}{5} = 0.2.\]
03
Calculate the Second Term
Substitute \( n = 2 \) into the sequence formula:\[a_2 = \frac{2^2}{2 \times 2 + 3 \times 2^2} = \frac{4}{4 + 12} = \frac{4}{16} = 0.25.\]
04
Calculate the Third Term
Substitute \( n = 3 \) into the sequence formula:\[a_3 = \frac{3^2}{2 \times 3 + 3 \times 3^2} = \frac{9}{6 + 27} = \frac{9}{33} \approx 0.2727.\]
05
Calculate the Subsequent Terms
Continue substituting values from \( n = 4 \) to \( n = 10 \) into the formula to find additional terms:- \( a_4 = \frac{16}{8 + 48} = \frac{16}{56} \approx 0.2857 \).- \( a_5 = \frac{25}{10 + 75} = \frac{25}{85} \approx 0.2941 \).- \( a_6 = \frac{36}{12 + 108} = \frac{36}{120} = 0.3 \).- \( a_7 = \frac{49}{14 + 147} = \frac{49}{161} \approx 0.3043 \).- \( a_8 = \frac{64}{16 + 192} = \frac{64}{208} \approx 0.3077 \).- \( a_9 = \frac{81}{18 + 243} = \frac{81}{261} \approx 0.3103 \).- \( a_{10} = \frac{100}{20 + 300} = \frac{100}{320} = 0.3125 \).
06
Plot the Terms on a Graph
Plot the calculated terms (\( n, a_n \)) from \( n = 1 \) to \( n = 10 \) on a coordinate plane to visualize the sequence.For instance: - Point for \( n = 1 \): \( (1, 0.2) \)- Point for \( n = 2 \): \( (2, 0.25) \)- Continue plotting until \( n = 10 \). Observe the trend in the terms as \( n \) increases.
07
Analyze the Limit of the Sequence
As \( n \) increases, the sequence values approach a steady state. In the sequence formula \( a_n = \frac{n^2}{2n + 3n^2} \), as \( n \) becomes very large, the term \( 3n^2 \) dominates both the numerator and the denominator:\[\lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} \frac{n^2}{3n^2} = \lim_{{n \to \infty}} \frac{1}{3} = \frac{1}{3}.\]Hence, the sequence converges to the limit \( \frac{1}{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Sequences
A sequence in mathematics is a list of numbers in a specific order, where each number is associated with a particular position, often described by variables like \(n\). Sequences are an essential part of calculus and mathematical analysis.
When you deal with sequences, you're looking at a function with a domain of natural numbers. Each term in a sequence is determined by its position and follows a specific formula, such as \(a_n = \frac{n^2}{2n + 3n^2}\).
When you deal with sequences, you're looking at a function with a domain of natural numbers. Each term in a sequence is determined by its position and follows a specific formula, such as \(a_n = \frac{n^2}{2n + 3n^2}\).
- The terms are generated by substituting consecutive natural numbers into the formula.
- It's important to examine the behavior of the sequence over different positions, starting from \(n = 1\).
The Concept of Limits
Limits allow us to understand the behavior of sequences as the terms approach infinity. In other words, it helps us determine what value a sequence might converge to as the number of terms increases.
When examining the limit of a sequence, you're often interested in what happens as \( n \) becomes very large. For example, with the sequence \(a_n = \frac{n^2}{2n + 3n^2}\), you want to see what the sequence approaches as \(n\) grows larger.
When examining the limit of a sequence, you're often interested in what happens as \( n \) becomes very large. For example, with the sequence \(a_n = \frac{n^2}{2n + 3n^2}\), you want to see what the sequence approaches as \(n\) grows larger.
- This is done by focusing on the dominant terms in the formula, both in the numerator and denominator.
- As \( n \rightarrow \infty \), irrelevant smaller terms can usually be ignored, simplifying the limit calculation.
Understanding Convergence
Convergence in mathematics refers to the idea of a sequence approaching a specific value, called the limit, as the sequence progresses. When we say a sequence converges, we're saying it "settles down" to a particular value as \( n \) becomes very large.
For the given sequence \( a_n = \frac{n^2}{2n + 3n^2} \), we determined that as \( n \) becomes large, the sequence converges to \( \frac{1}{3} \). Here's how:
For the given sequence \( a_n = \frac{n^2}{2n + 3n^2} \), we determined that as \( n \) becomes large, the sequence converges to \( \frac{1}{3} \). Here's how:
- The term \( 3n^2 \) heavily influences both the numerator and denominator.
- After simplifying the formula in the limit, it was shown that the sequence converges to \( \frac{1}{3} \).
Exploring Mathematical Analysis
Mathematical analysis helps us understand the underlying principles behind functions, sequences, and limits. It involves rigorously studying and solving a wide array of mathematical problems, often using calculus techniques.
Through analysis, we can:
Through analysis, we can:
- Delve deeply into concepts like sequences and their behaviors.
- Explore how limits define and describe infinite processes.
- Assess convergence and divergence of sequences and series.
