Problem 7 - College Algebra

Question

Write the first six terms of each arithmetic sequence. $$a_{1}=\frac{5}{2}, d=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are \(a_{1}=\frac{5}{2}\), \(a_{2}=2\), \(a_{3}=\frac{3}{2}\), \(a_{4}=1\), \(a_{5}=\frac{1}{2}\), \(a_{6}=0\)

Step 1: Identify First Term \(a_{1}\) and Common Difference d

From the given, the first term \(a_{1}\) of the arithmetic sequence is \(\frac{5}{2}\) and the common difference, d, is -\(\frac{1}{2}\).

Step 2: Find the Second Term \(a_{2}\)

Using the arithmetic sequence formula \(a_{n} = a_{1} + (n-1) \cdot d\), plug in \(a_{1} = \frac{5}{2}\), d = -\(\frac{1}{2}\), and n = 2 to find the second term. So, \(a_{2} = \frac{5}{2} + (2-1) \cdot -\frac{1}{2}= 2\)

Step 3: Find the Third Term \(a_{3}\)

Again use the formula, but this time with n = 3. \(a_{3} = \frac{5}{2} + (3-1) \cdot -\frac{1}{2}= \frac{3}{2}\)

Step 4: Find the Fourth Term \(a_{4}\)

Apply the formula with n = 4. \(a_{4} = \frac{5}{2} + (4-1) \cdot -\frac{1}{2}= 1\)

Step 5: Find the Fifth Term \(a_{5}\)

Apply the formula with n = 5. \(a_{5} = \frac{5}{2} + (5-1) \cdot -\frac{1}{2}= \frac{1}{2}\)

Step 6: Find the Sixth Term \(a_{6}\)

Apply the formula with n = 6. \(a_{6} = \frac{5}{2} + (6-1) \cdot -\frac{1}{2}= 0\)