Problem 11 - Algebra for College Students

Question

Find the required term of each of the sequences. The 19 th term of \(1,5,9,13, \ldots\) 73

Step-by-Step Solution

Answer

The 19th term of the sequence is 73.

Step 1: Identify the sequence pattern

Examine the sequence given: 1, 5, 9, 13, ... and notice that each term increases by 4 from the previous term. This indicates that the sequence is arithmetic with a common difference, denoted as \( d = 4 \).

Step 2: Confirm the formula of an arithmetic sequence

The formula to find the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 3: Plug values into the formula

In this problem, the first term \( a_1 = 1 \), the common difference \( d = 4 \), and the term number \( n = 19 \). Substitute these values into the formula: \( a_{19} = 1 + (19-1) \cdot 4 \).

Step 4: Simplify the expression

Calculate the expression: \( a_{19} = 1 + 18 \cdot 4 \). First, multiply \( 18 \cdot 4 \) to get 72, then add 1 to obtain \( a_{19} = 73 \).

Key Concepts

Common Difference

In the world of sequences, understanding the 'common difference' is crucial when dealing with arithmetic sequences. An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This consistent gap is what we call the 'common difference'. For instance, consider the sequence 1, 5, 9, 13, ... Here, as you move from one term to the next, each time you add 4. Hence, the common difference ( d ) is 4.
This uniformity simplifies predictions and calculations within the sequence, as it forms the backbone of how the sequence progresses. Recognizing and calculating the common difference is the first step in solving many problems related to arithmetic sequences.

• To identify the common difference, subtract any term from the previous term.
• Ensure consistency by confirming the same difference across multiple pairs of consecutive terms.

This foundational concept makes it easier to apply formulas and compute terms quickly.

Arithmetic Sequence Formula

Once you understand the common difference, you can use the arithmetic sequence formula. This formula allows us to find any term in the sequence without listing all of the previous terms. The formula is:
\[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( n \) is the position of the term you are finding.
• \( d \) is the common difference.

This formula offers a direct approach to pinpoint any position in the sequence. By simply plugging in known values from your sequence into the formula, you can find your desired term efficiently.
The formula's simplicity is the key benefit, making arithmetic sequences more manageable. Just remember: always double-check your inputs to ensure accuracy in your calculations.

Nth Term Calculation

Calculating the nth term in an arithmetic sequence can save you time, especially with sequences comprising large numbers or when seeking terms far along the sequence. Let’s break it down using the sequence 1, 5, 9, 13, ... Here’s how you find the 19th term:

• First, identify \( a_1 = 1 \), \( d = 4 \), and \( n = 19 \). These come from the sequence and your target term.
• Insert these values into the arithmetic sequence formula: \( a_{19} = 1 + (19-1) \cdot 4 \).
• Solve inside the brackets: \( 19-1 = 18 \).
• Multiply by the common difference: \( 18 \cdot 4 = 72 \).
• Add \( a_1 \) to this product: \( 1 + 72 = 73 \).

Thus, the 19th term is 73. Through this method, you directly reach the position required without unnecessary calculations or guesswork.
This approach is ideal for efficiency and accuracy, allowing you to tackle any position in the sequence confidently.