Problem 34 - Precalculus with Limits

Question

In Exercises 33 - 40, write the first five terms of the arithmetic sequence. \( a_1 = 5, d = -\dfrac{3}{4} \)

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence are: \(5, 4.25, 3.5, 2.75, 2\)

Step 1: Determine Second Term

First, let's find out the second term \( a_2 \). Apply the general formula \( a_n = a_1 + (n-1) * d \). Replacing \( n = 2 \), \( a_1 = 5 \) and \( d = -\dfrac{3}{4} \), we get \( a_2 = 5 + (2-1) * -\dfrac{3}{4} = 5 - \dfrac{3}{4} = \dfrac{20}{4} - \dfrac{3}{4} = \dfrac{17}{4} \) which simplifies to \( a_2 = 4.25 \)

Step 2: Determine Third Term

Apply the same formula to find the third term \( a_3 \). Replace \( n = 3 \), \( a_1 = 5 \) and \( d = -\dfrac{3}{4} \) in the general formula to get \( a_3 = 5 + (3-1) * -\dfrac{3}{4} = 5 - \dfrac{6}{4} = \dfrac{20}{4} - \dfrac{6}{4} = \dfrac{14}{4} \) which simplifies to \( a_3 = 3.5 \).

Step 3: Determine Fourth Term

For the fourth term \( a_4 \), replace \( n = 4 \), \( a_1 = 5 \) and \( d = -\dfrac{3}{4} \) in the general formula to get \( a_4 = 5 + (4-1) * -\dfrac{3}{4} = 5 - \dfrac{9}{4} = \dfrac{20}{4} - \dfrac{9}{4} = \dfrac{11}{4} \) which simplifies to \( a_4 = 2.75 \).

Step 4: Determine Fifth Term

Finally, for the fifth term \( a_5 \), replace \( n = 5 \), \( a_1 = 5 \) and \( d = -\dfrac{3}{4} \) in the general formula to get \( a_5 = 5 + (5-1) * -\dfrac{3}{4} = 5 - \dfrac{12}{4} = \dfrac{20}{4} - \dfrac{12}{4} = \dfrac{8}{4} \) which simplifies to \( a_5 = 2 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element. It represents the consistent amount that is added or subtracted to each term to get the next term in the sequence.
This concept is central because it defines the sequence. Understanding the common difference is key to grasping the entire sequence.

• The common difference is denoted by "d".
• In an arithmetic sequence, the difference between consecutive terms is always the same.
• To find the common difference, subtract any term from the term that follows it.

For instance, in this problem with a sequence starting at 5 and with a common difference of \(-\frac{3}{4}\), means each term decreases by \(\frac{3}{4}\) from the previous one.

General Formula of Arithmetic Sequence

The general formula for an arithmetic sequence is your tool to find any term in the sequence without listing all the previous terms. This formula is useful, especially when dealing with large indices.
The general formula is:

• \(a_n = a_1 + (n-1) \times d\)
• Here, \(a_n\) is the "n"th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Using this formula helps save time and avoid mistakes by solving directly for any term. For example, by plugging in the values \(a_1 = 5\), \(d = -\frac{3}{4}\), and \(n = 2\), we calculated the second term as \(4.25\). This systematic approach can be repeated for any term.

Term Calculation

Term calculation in an arithmetic sequence is straightforward once you know the first term and the common difference. Each subsequent term depends directly on the previous terms.
This systematic procedure ensures each term is accurately determined:

• First, plug your known values into the general formula.
• Perform the arithmetic operations carefully, especially when dealing with fractions or negative numbers.
• This will give you the correct term within the sequence.

For our exercise, we calculated the terms as follows:

• \(a_2 = 4.25\)
• \(a_3 = 3.5\)
• \(a_4 = 2.75\)
• \(a_5 = 2\)

Executing these steps methodically ensures each term's accuracy, showcasing the beauty of simplicity and consistency within arithmetic sequences.