91Ó°ÊÓ

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.

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.

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.