Problem 4 - Precalculus: Mathematics for Calculus

Question

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=2 n-1\)

Step-by-Step Solution

Answer

The first four terms are 1, 3, 5, and 7. The 100th term is 199.

Step 1: Identify the general formula

The general formula for the sequence is given by \( a_n = 2n - 1 \). This formula allows us to determine any term in the sequence based on its position, \( n \).

Step 2: Find the first term

To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \( a_1 = 2(1) - 1 = 1 \). Thus, the first term is 1.

Step 3: Find the second term

For the second term \( a_2 \), substitute \( n = 2 \) into the formula: \( a_2 = 2(2) - 1 = 3 \). Thus, the second term is 3.

Step 4: Find the third term

To find the third term \( a_3 \), substitute \( n = 3 \) into the formula: \( a_3 = 2(3) - 1 = 5 \). Thus, the third term is 5.

Step 5: Find the fourth term

For the fourth term \( a_4 \), substitute \( n = 4 \) into the formula: \( a_4 = 2(4) - 1 = 7 \). Thus, the fourth term is 7.

Step 6: Find the 100th term

To find the 100th term \( a_{100} \), substitute \( n = 100 \) into the formula: \( a_{100} = 2(100) - 1 = 199 \). Thus, the 100th term is 199.

Key Concepts

Sequence Formula

In arithmetic sequences, a sequence formula is used to find any term of the sequence without needing to list all its previous terms. A common sequence formula involves terms increasing by a constant difference. The sequence in our exercise is defined by the formula \( a_n = 2n - 1 \). This formula is expressed in terms of \( n \), which represents the position of the term in the sequence. By substituting any positive integer for \( n \), you can calculate the corresponding term.

• For example, when \( n = 1 \), \( a_1 = 2(1) - 1 = 1 \).
• This formula highlights that the sequence increases linearly, which is characteristic of arithmetic sequences.

Understanding how to use this formula simplifies the process of finding terms in the sequence, as well as exploring patterns without manually computing each term.

General Term

The general term of a sequence is a formula that enables you to find any specific term without deriving all preceding terms. In our particular case, the general formula \( a_n = 2n - 1 \) gives us the nth term.

• If you are tasked with finding the 100th term, substitute \( n = 100 \) into the formula: \( a_{100} = 2(100) - 1 = 199 \).
• This efficiently provides us with the term at any position \( n \) in the sequence.

Using the general term, one can immediately determine any term, which is especially helpful when dealing with large numbered terms, like the 100th term. Moreover, it underscores the predictable and recurring nature of arithmetic sequences, making them easier to analyze and understand.

Algebra

Algebra plays a foundational role in understanding sequences and formulas. It allows us to express relationships and patterns using symbols and variables in a structured manner. In the sequence \( a_n = 2n - 1 \), algebra provides a compact way to define an infinite series of numbers:

• It uses \( n \), a variable, to elements in the sequence positions.
• Through simple operations, such as multiplication and subtraction (where \( 2n - 1 \) calculates terms), algebra simplifies complex sequences into manageable equations.

By mastering algebraic expressions and their manipulation, students gain the ability to define and calculate any term within a sequence quickly and efficiently. Algebraic techniques are crucial for recognizing patterns within sequences and solving higher-level mathematical problems with ease.