Problem 28 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(1,3,5,7, \dots\)

Step-by-Step Solution

Answer

The nth term of the sequence is \( 2n - 1 \).

Step 1: Identify the Pattern

To find the pattern, examine the differences between consecutive terms. The first term is 1, the second term is 3, the third term is 5, and the fourth term is 7. Calculate the difference for each consecutive pair: - From 1 to 3, the difference is 2. - From 3 to 5, the difference is 2. - From 5 to 7, the difference is 2. This indicates that the sequence increases by 2 at each step.

Step 2: Determine the General Term Formula

Since the sequence increases by 2 each time, it is an arithmetic sequence. The formula for the nth term of an arithmetic sequence is given by:\[ a_n = a_1 + (n-1) imes d \]where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Here, \( a_1 = 1 \) and \( d = 2 \). Substitute these into the formula:\[ a_n = 1 + (n-1) imes 2 \]

Step 3: Simplify the Formula

Simplify the expression to get the general term:\[ a_n = 1 + (n-1) imes 2 \]Distribute the 2:\[ a_n = 1 + 2n - 2 \]Combine like terms:\[ a_n = 2n - 1 \]Thus, the nth term of the sequence is \( 2n - 1 \).

Key Concepts

Understanding the nth Term

Every sequence consists of multiple terms where each specific position of a term within the sequence is called the 'nth term'. For example, in a sequence like 2, 4, 6, 8,... identifying the 5th term requires understanding the position of the term in question relative to the sequence. The nth term plays a critical role because it allows us to calculate any term's value without needing to know all preceding terms. This makes sequences predictable and easier to analyze, especially in arithmetic sequences where each step follows a repetitive pattern. In the case of our sequence 1, 3, 5, 7, ..., the nth term will allow us to find any term we wish, such as the 100th term, simply by plugging in 100 for the value of n into our formula.

Defining Common Difference

The common difference in an arithmetic sequence is what sets the terms apart from each other, establishing a steady, predictable pattern. The common difference is the amount added to (or subtracted from, if negative) each term to reach the subsequent term. In the sequence 1, 3, 5, 7,..., we notice:

• The difference between 1 and 3 is 2.
• Between 3 and 5 is also 2.
• Similarly, between 5 and 7 is 2 as well.

This uniformity tells us our common difference, denoted by 'd', is 2. Understanding this helps hash out the specific behavior of our sequence. In any arithmetic sequence, the constant common difference means the interval between consecutive terms remains unchanged which is key to forming the formula for finding the nth term.

Establishing the General Term Formula

The cornerstone of arithmetic sequences is their general term formula. This all-encompassing formula gives a straightforward way to figure out any term’s value using its position in the sequence. The general term formula is denoted as:\[a_n = a_1 + (n-1) \times d\]

• \(a_n\) is the nth term we wish to find.
• \(a_1\) represents the first term of the sequence.
• \(n\) is the term number.
• and \(d\) is the common difference.

Applying this to our sequence, 1, 3, 5, 7,..., where \(a_1 = 1\) and \(d = 2\), we input these into the formula to find\(a_n\):\[a_n = 1 + (n-1) \times 2\]Simplifying provides us with \(a_n = 2n - 1\). This formula now allows calculation of any nth term. For instance, setting \(n = 5\), you'd find the 5th term as: \(10 - 1 = 9\). This method is powerful because it converts a seemingly complex task of identifying terms into simple arithmetic operations.