Problem 3 - Algebra and Trigonometry

Question

1–4 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). $$a_{n}=\frac{5}{2}-(n-1)$$

Step-by-Step Solution

Answer

First five terms: \(\frac{5}{2}, \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}\); Common difference \(d = -1\).

Step 1: Find the First Term

To find the first term of the sequence, substitute \(n = 1\) into the sequence formula \(a_{n} = \frac{5}{2} - (n-1)\).\[ a_{1} = \frac{5}{2} - (1-1) = \frac{5}{2} \]. Thus, the first term is \(\frac{5}{2}\).

Step 2: Calculate Subsequent Terms

Use the sequence formula to find the next four terms by substituting \(n = 2\), \(n = 3\), \(n = 4\), and \(n = 5\). For each \(n\):- \(a_{2} = \frac{5}{2} - (2-1) = \frac{5}{2} - 1 = \frac{3}{2}\)- \(a_{3} = \frac{5}{2} - (3-1) = \frac{5}{2} - 2 = \frac{1}{2}\)- \(a_{4} = \frac{5}{2} - (4-1) = \frac{5}{2} - 3 = -\frac{1}{2}\)- \(a_{5} = \frac{5}{2} - (5-1) = \frac{5}{2} - 4 = -\frac{3}{2}\).These terms are \(\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}\).

Step 3: Determine the Common Difference

The common difference \(d\) in an arithmetic sequence is found by subtracting any term from the previous term. Using the first two terms:\[ d = a_{2} - a_{1} = \frac{3}{2} - \frac{5}{2} = -1\]. Thus, the common difference is \(-1\).

Step 4: Graph the Sequence

Plot the terms from step 1 and 2 on the Cartesian plane, where the x-axis represents the position in the sequence (1 for \(a_1\), 2 for \(a_2\), etc.) and the y-axis represents the value of the terms. The points to plot are (1, \(\frac{5}{2}\)), (2, \(\frac{3}{2}\)), (3, \(\frac{1}{2}\)), (4, \(-\frac{1}{2}\)), and (5, \(-\frac{3}{2}\)). Connect these points to show the linear trend in the sequence.

Key Concepts

Common Difference

At the heart of every arithmetic sequence is the common difference. This is the consistent amount added or subtracted from each term to get the next. In our exercise, we have the sequence defined by the formula \( a_{n} = \frac{5}{2} - (n-1) \).To find the common difference \( d \), we subtract any term from the one that precedes it. In this case, \( a_{2} = \frac{3}{2} \) and \( a_{1} = \frac{5}{2} \). The calculation shows: \[ d = a_{2} - a_{1} = \frac{3}{2} - \frac{5}{2} = -1 \]This indicates that our sequence decreases by 1 with each step. Understanding the common difference is crucial, as it helps predict any term in the series quickly by using the formula:

• \( a_{n} = a_1 + (n-1)d \)

By knowing \( a_1 = \frac{5}{2} \) and \( d = -1 \), you can easily calculate any future term in the sequence.

Sequence Graphing

Graphing an arithmetic sequence visually demonstrates its uniform change. Let's map our sequence with the terms \( \frac{5}{2}, \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \). Each term's position becomes a point on the Cartesian plane

• (1, \( \frac{5}{2} \))
• (2, \( \frac{3}{2} \))
• (3, \( \frac{1}{2} \))
• (4, \( -\frac{1}{2} \))
• (5, \( -\frac{3}{2} \))

On the graph, the x-axis shows the position in the sequence, while the y-axis shows the term's value.
Link these points with a straight line to observe the sequence's linear pattern. It's important as it helps visualize the predictability of linear sequences. Notice that as you move one position right (increase \( n \)), the line descends one unit, reflecting our common difference of \( -1 \). Thus, the graph reinforces the calculation of each subsequent term.

First Five Terms

Finding the first five terms of an arithmetic sequence involves direct substitution into the sequence formula. For our formula, \( a_{n} = \frac{5}{2} - (n-1) \), we substitute successively increasing values for \( n \) starting from 1.Let's calculate:

• For \( n = 1 \), \( a_1 = \frac{5}{2} \)
• For \( n = 2 \), \( a_2 = \frac{3}{2} \)
• For \( n = 3 \), \( a_3 = \frac{1}{2} \)
• For \( n = 4 \), \( a_4 = -\frac{1}{2} \)
• For \( n = 5 \), \( a_5 = -\frac{3}{2} \)

These numbers are the starting points of our arithmetic sequence, demonstrating how it steadily decreases by the common difference \( d = -1 \). Understanding these initial terms is key to grasping the fundamental rules and flow of arithmetic sequences.