Problem 10 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = 4n - 7 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are -3, 1, 5, 9, 13.

Step 1: Understand the sequence

The sequence in the exercise is described by the function \( a_n = 4n - 7 \), where \( n \) is the position of the term in the sequence starting from 1.

Step 2: Substitute n = 1 into the function

To find the first term of the sequence, substitute \( n = 1 \) into the function. Hence, \( a_1 = 4(1) - 7 = -3 \). So, the first term of the sequence is -3.

Step 3: Substitute n = 2 into the function

To find the second term, substitute \( n = 2 \) into the function. Thus, \( a_2 = 4(2) - 7 = 1 \). So, the second term of the sequence is 1.

Step 4: Substitute n = 3 into the function

To find the third term, substitute \( n = 3 \) into the function. We get \( a_3 = 4(3) - 7 = 5 \). So, the third term of the sequence is 5.

Step 5: Substitute n = 4 into the function

To find the fourth term, substitute \( n = 4 \) into the function. Hence, \( a_4 = 4(4) - 7 = 9 \). So, the fourth term of the sequence is 9.

Step 6: Substitute n = 5 into the function

To find the fifth term, substitute \( n = 5 \) into the function. Thus, \( a_5 = 4(5) - 7 = 13 \). So, the fifth term of the sequence is 13.

Key Concepts

Sequence Terms Calculation

Understanding the calculation of terms in a sequence is fundamental when exploring mathematical patterns. In sequences, each term is defined by its position, and for arithmetic sequences, we follow a specific arithmetic rule to calculate terms. Given the function provided in the exercise, \( a_n = 4n - 7 \), we calculate the terms by substituting the term's position number for \(n\). For instance, to find the third term (\(a_3\)), we replace \(n\) with 3, resulting in \(a_3 = 4(3) - 7 = 5\).

This process is repeated for as many terms as required, consistently applying the given function to each term's position number. By doing so, we build a sequence where the relationship between consecutive terms becomes clearer with each step. Remember, careful substitution and following arithmetic rules are keys to accurate sequence term calculation.

Arithmetic Sequence Formula

An arithmetic sequence is characterized by a constant difference between consecutive terms, referred to as the 'common difference'. The formula for any term \( a_n \) in an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

In our example sequence, we can derive that the common difference \( d \) is 4, as it is the coefficient that multiplies the term's position number \( n \) in the formula. The formula simplifies the calculation of any term in the sequence without the need to calculate all the previous terms – a shortcut that saves time and makes understanding mathematical sequences in precalculus easier.

Mathematical Sequences in Precalculus

Precalculus is often the introduction to mathematical sequences, laying the foundation for more advanced topics in calculus. Sequences are ordered lists of numbers where each number is defined by a specific rule. They come in various types, such as arithmetic, geometric, or more complex recursive sequences. Understanding sequences in precalculus involves recognizing these patterns and applying formulas to find sequence terms or analyze their behavior.

An essential part of studying sequences in precalculus is to understand their properties, such as monotonicity, boundedness, and convergence. Grasping these concepts allows students to solve problems efficiently and transition smoothly into calculus, where sequences play a pivotal role in defining limits and series.

Understanding Sequences in Math

Sequences in mathematics are not just lists of numbers but are structured in such a way that there is a definite pattern that dictates the formation of successive terms. Understanding sequences is pivotal as they form the basis for series and various mathematical analyses. Fundamental to this understanding is grasping the general term, or the nth term, which is a formula that expresses the sequence's rule.

When learned effectively, sequences offer a glimpse into the beauty and order inherent in mathematics. Students are encouraged to look beyond the calculations and see the broader applications, such as modeling real-world phenomena and solving practical problems. Developing intuitive knowledge of sequences and their properties equips students with the tools to tackle a wide array of mathematical challenges.