Problem 30 - College Algebra with Modeling and Visualization

Question

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$ 4,1,-2,-5,-8 $$

Step-by-Step Solution

Answer

The numerical representation is 4,1,-2,-5,-8,-11,-14,-17. Graphically, plot these points and connect them. Symbolically, the sequence is given by \( a_n = -3n + 7 \).

Step 1: Identify the First Term of the Sequence

The first term \( a_1 \) of the sequence is given directly as the first number in the sequence, which is 4.

Step 2: Determine the Common Difference

To find the common difference \( d \) of an arithmetic sequence, subtract the first term from the second term. This gives: \( d = 1 - 4 = -3 \). Verify this is consistent for subsequent pairs: \( (-2) - 1 = -3 \), \( (-5) - (-2) = -3 \), confirming that the common difference is \( -3 \).

Step 3: Generate More Terms of the Sequence

Using the formula for the nth term of an arithmetic sequence, \( a_n = a_1 + (n-1)d \), calculate the next three terms after -8: \( a_6 = 4 + 5(-3) = -11 \), \( a_7 = 4 + 6(-3) = -14 \), and \( a_8 = 4 + 7(-3) = -17 \).

Step 4: Numerical Representation

The numerical representation of at least eight terms of the sequence is: \( 4, 1, -2, -5, -8, -11, -14, -17 \).

Step 5: Graphical Representation

Plot each term of the sequence on a coordinate plane where the x-axis represents the term number (n) and the y-axis represents the value of each term. Plot the points: (1,4), (2,1), (3,-2), (4,-5), (5,-8), (6,-11), (7,-14), (8,-17). Connect the points to show the linear trend of the sequence.

Step 6: Symbolic Representation

The symbolic representation of the arithmetic sequence is the equation \( a_n = 4 + (n-1)(-3) \), or simplified, \( a_n = -3n + 7 \). This formula can generate any term in the sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the consistent interval between consecutive terms. This difference gives the sequence its linear pattern. In the given sequence, start with the first two terms. Subtract the first term, 4, from the next term, 1, to determine the common difference: \[ d = 1 - 4 = -3 \] This means every next number in the series is 3 units less than the preceding one. To ensure the correctness of this common difference, verify it.

• The third term, -2, minus the second term, 1, is also -3.
• Continue this process with the fourth term: \[ -5 - (-2) = -3 \]
• And fifth term: \[ -8 - (-5) = -3 \]

In each case, the result is consistent, confirming that the common difference for this sequence is \(-3\).

Nth Term Formula

The formula for the nth term of an arithmetic sequence lets us calculate any term in the sequence readily. This formula is given by\[ a_n = a_1 + (n-1)d \]where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. In our sequence, the first term \(a_1\) is 4, and the common difference \(d\) is \(-3\). Using these values, the formula becomes:\[ a_n = 4 + (n-1)(-3) \]This can be simplified as:\[ a_n = -3n + 7 \] With this formula, you can generate any term in the sequence by plugging in the desired term number for \(n\). For example, to find the 8th term:\[ a_8 = -3(8) + 7 = -24 + 7 = -17 \] Each term uses the same straightforward calculation.

Graphical Representation

Graphically representing an arithmetic sequence involves plotting each term on a coordinate plane. The x-axis signifies the term number, and the y-axis denotes the term's value. Using the formula \( a_n = -3n + 7 \), calculate and plot at least eight terms of the sequence:

• 1st term: (1, 4)
• 2nd term: (2, 1)
• 3rd term: (3, -2)
• 4th term: (4, -5)
• 5th term: (5, -8)
• 6th term: (6, -11)
• 7th term: (7, -14)
• 8th term: (8, -17)

Once plotted, connect the points with a straight line. This straight line visually displays the linear nature of the arithmetic sequence and provides insight into the relationship between the term number and the term's value.

Numerical Representation

Numerical representation conveys the sequence as a series of values without any graphical display or symbolic formulation. In this arithmetic sequence, the numerical series begins with \(4\) and continues according to the common difference, \(-3\). List out at least eight terms to represent the sequence numerically:

• 1st term: 4
• 2nd term: 1
• 3rd term: -2
• 4th term: -5
• 5th term: -8
• 6th term: -11
• 7th term: -14
• 8th term: -17

This representation helps in understanding the pattern of the sequence directly through its values. Each consecutive term is clear, demonstrating the sequence's regular pattern.

Symbolic Representation

The symbolic representation of an arithmetic sequence is a concise expression that describes every term in the sequence. This format captures the essence of the sequence using an equation. In this case, the symbolic equation from our sequence's nth term formula is:\[ a_n = -3n + 7 \] This equation allows anyone to calculate any term by substituting \(n\) with the specific term number. This representation simplifies sequence advancement and helps map the behavior of the sequence over large numbers of terms. It's a versatile form, providing both theoretical insights and practical calculations.