Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{4 n-1}{n^{2}+2}$$

Step-by-Step Solution

Answer

The first five terms are 1, \(\frac{7}{6}\), 1, \(\frac{5}{6}\), and \(\frac{19}{27}\).

Step 1: Define the Formula for the Sequence

We have the formula for the sequence: \( a_n = \frac{4n - 1}{n^2 + 2} \). Our task is to find the first five terms of this sequence, meaning we have to evaluate the sequence formula for \( n = 1, 2, 3, 4, \text{ and } 5 \).

Step 2: Calculate the First Term (n=1)

Substitute \( n = 1 \) into the sequence formula:\[ a_1 = \frac{4(1) - 1}{1^2 + 2} = \frac{4 - 1}{1 + 2} = \frac{3}{3} = 1. \]Therefore, the first term is \( a_1 = 1 \).

Step 3: Calculate the Second Term (n=2)

Substitute \( n = 2 \) into the sequence formula:\[ a_2 = \frac{4(2) - 1}{2^2 + 2} = \frac{8 - 1}{4 + 2} = \frac{7}{6}. \]Thus, the second term is \( a_2 = \frac{7}{6} \).

Step 4: Calculate the Third Term (n=3)

Substitute \( n = 3 \) into the sequence formula:\[ a_3 = \frac{4(3) - 1}{3^2 + 2} = \frac{12 - 1}{9 + 2} = \frac{11}{11} = 1. \]Hence, the third term is \( a_3 = 1 \).

Step 5: Calculate the Fourth Term (n=4)

Substitute \( n = 4 \) into the sequence formula:\[ a_4 = \frac{4(4) - 1}{4^2 + 2} = \frac{16 - 1}{16 + 2} = \frac{15}{18} = \frac{5}{6}. \]Therefore, the fourth term is \( a_4 = \frac{5}{6} \).

Step 6: Calculate the Fifth Term (n=5)

Substitute \( n = 5 \) into the sequence formula:\[ a_5 = \frac{4(5) - 1}{5^2 + 2} = \frac{20 - 1}{25 + 2} = \frac{19}{27}. \]Thus, the fifth term is \( a_5 = \frac{19}{27} \).

Key Concepts

Sequence Formula

Understanding the concept of a sequence formula is crucial in precalculus. A sequence formula is essentially a mathematical blueprint that generates the terms in a sequence. It allows us to calculate each term in a sequence by plugging in different values for the variable, typically denoted by \( n \).
For the given exercise, we have the sequence formula:

• \( a_n = \frac{4n - 1}{n^2 + 2} \)

This sequence formula defines a rule for the numerical sequence. The variable \( n \) typically starts at a specific integer and increases consecutively. In our example, \( n \) takes on the values 1, 2, 3, etc., and the formula calculates the sequence's terms based on these values.
The sequence formula is an alternative to listing each term manually and exemplifies how sequences evolve mathematically.

Sequence Terms

After identifying the sequence formula, the next step is understanding sequence terms. Sequence terms refer to the individual elements generated by applying the sequence rule to particular values of \( n \). In simple terms, these are the results we obtain by substituting values into the sequence formula.
For example, in this particular sequence, the first five sequence terms are computed as follows:

• First term, \( a_1 = 1 \)
• Second term, \( a_2 = \frac{7}{6} \)
• Third term, \( a_3 = 1 \)
• Fourth term, \( a_4 = \frac{5}{6} \)
• Fifth term, \( a_5 = \frac{19}{27} \)

The beauty of sequence terms is in their predictability; following the sequence formula produces consistent results corresponding to each consecutive value of \( n \). Each term is distinct, yet part of a collective series.

Sequence Evaluation

Once we have the sequence formula and begin to identify sequence terms, we move on to sequence evaluation. This involves the actual calculation process of plugging values into the formula and simplifying them.
In the case of our sequence formula \( a_n = \frac{4n - 1}{n^2 + 2} \), evaluating the sequence requires substituting integers for \( n \) and performing algebraic operations:

• For \( n = 1 \), calculate \( a_1 = \frac{3}{3} = 1 \).
• For \( n = 2 \), calculate \( a_2 = \frac{7}{6} \).
• For \( n = 3 \), calculate \( a_3 = \frac{11}{11} = 1 \).
• For \( n = 4 \), calculate \( a_4 = \frac{5}{6} \).
• For \( n = 5 \), calculate \( a_5 = \frac{19}{27} \).

The evaluation process involves arithmetic and division. For some terms, additional simplification such as reducing fractions might be required. Sequence evaluation not only demands attention to detail but also encourages thinking analytically to ensure each computed term fits the pattern established by the sequence formula.