Problem 6 - Algebra and Trigonometry

Question

\(5-8=\) A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=3-4(n-1) $$

Step-by-Step Solution

Answer

The first five terms are 3, -1, -5, -9, -13, with a common difference of -4.

Step 1: Identify Sequence Formula

The given sequence is described by the formula \( a_{n} = 3 - 4(n-1) \). This formula is an arithmetic sequence where \( n \) is the term number.

Step 2: Find the First Term

To find the first term \( a_1 \), substitute \( n = 1 \) into the sequence formula: \( a_1 = 3 - 4(1-1) = 3 - 4(0) = 3 \). Therefore, the first term is 3.

Step 3: Calculate the Next Four Terms

Find the subsequent terms by substituting \( n \) values 2, 3, 4, and 5 into the formula:\[ a_2 = 3 - 4(2-1) = 3 - 4 = -1 \] \[ a_3 = 3 - 4(3-1) = 3 - 8 = -5 \] \[ a_4 = 3 - 4(4-1) = 3 - 12 = -9 \] \[ a_5 = 3 - 4(5-1) = 3 - 16 = -13 \] Thus, the first five terms are 3, -1, -5, -9, -13.

Step 4: Determine the Common Difference (d)

The common difference \( d \) in an arithmetic sequence is found by subtracting the first term from the second term. Thus, \( d = a_2 - a_1 = -1 - 3 = -4 \). The common difference is -4.

Step 5: Graph the Terms

To graph the terms, plot the term number \( n \) on the x-axis and the corresponding term \( a_n \) on the y-axis. Plot the points: (1, 3), (2, -1), (3, -5), (4, -9), (5, -13). Join these to see the linear pattern typical of an arithmetic sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key concept that defines how much the value of each term increases or decreases from the previous term. The common difference, denoted as \(d\), is consistent across the entire sequence, meaning every pair of consecutive terms differs by this amount.

• To find the common difference, subtract the first term from the second term.
• Mathematically, it is represented as \(d = a_{n+1} - a_n\), where \(a_n\) is the \(n\)-th term of the sequence.

In our sequence \(a_n = 3 - 4(n-1)\), you can calculate the common difference by taking any two consecutive terms. For example, the difference between the second term, -1, and the first term, 3, provides us with \(d = -1 - 3 = -4\). Thus, in this sequence, each term is 4 units less than the term preceding it.Understanding the common difference helps you easily determine future terms and recognize the sequence type as arithmetic.

Sequence Formula

A sequence formula, also known as the general term formula of a sequence, serves as a blueprint to find any term in the sequence without needing the previous terms. An arithmetic sequence formula employs a linear pattern given by \(a_n = a_1 + (n-1) \cdot d\), making it simple enough to compute any term you desire.

• \(a_1\) represents the first term in the sequence.
• \(n\) is the term number you are solving for.
• \(d\) stands for the common difference.

In our case, the sequence formula is \(a_n = 3 - 4(n-1)\). You can immediately plug in any value of \(n\) to determine the corresponding term \(a_n\). This formula essentially follows the pattern of the first term minus 4 times one less than the term number. Learning to interpret and manipulate the sequence formula is crucial for understanding how each term is derived, helping you grasp larger arithmetic concepts.

Graphing Sequences

Graphing an arithmetic sequence is a visual method to analyze the pattern and verify its linear nature. To create the graph:

• Mark the term numbers on the x-axis.
• Align the sequence values along the y-axis.
• Plot each (n, a_n) pair as a point in the coordinate plane.

In our example, you'd plot the points like this:- (1, 3) - (2, -1) - (3, -5)- (4, -9)- (5, -13)Once plotted, connect the dots in a straight line.
This straight line confirms the constant common difference \(d = -4\), demonstrating the linear downward slope expected in a decreasing arithmetic sequence.
Graphing aids visually in detecting errors in term calculations and provides a concrete overview of how the sequence behaves across different term settings.