Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. \(a_{1}=0, d=\frac{2}{3}\)

Step-by-Step Solution

Answer

The first five terms are 0, \(\frac{2}{3}\), \(\frac{4}{3}\), 2, \(\frac{8}{3}\).

Step 1: Understanding Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by \(d\). The first term of an arithmetic sequence is denoted by \(a_1\).

Step 2: Given values

We are provided with the first term \(a_{1}=0\) and the common difference \(d=\frac{2}{3}\).

Step 3: Applying the formula for terms

The formula for the \(n^{th}\) term of an arithmetic sequence is given by: \(a_n = a_1 + (n-1)d\). We will use this formula to find each of the first five terms.

Step 4: Calculating the first term \(a_1\)

The first term is already given as \(a_1 = 0\).

Step 5: Calculating the second term \(a_2\)

Using the formula: \(a_2 = a_1 + (2-1)d = 0 + 1 \times \frac{2}{3} = \frac{2}{3}\).

Step 6: Calculating the third term \(a_3\)

Using the formula: \(a_3 = a_1 + (3-1)d = 0 + 2 \times \frac{2}{3} = \frac{4}{3}\).

Step 7: Calculating the fourth term \(a_4\)

Using the formula: \(a_4 = a_1 + (4-1)d = 0 + 3 \times \frac{2}{3} = 2\).

Step 8: Calculating the fifth term \(a_5\)

Using the formula: \(a_5 = a_1 + (5-1)d = 0 + 4 \times \frac{2}{3} = \frac{8}{3}\).

Key Concepts

Common Difference

In an arithmetic sequence, the most vital part is understanding the role of the common difference. The common difference is the constant amount that you add (or subtract) to each term to get to the next term in the sequence. It's denoted by \(d\). In our exercise, the common difference is \(\frac{2}{3}\).

• If the common difference is positive, each term will be larger than the previous one, indicating a sequence that is increasing.
• If the common difference is negative, each term will be smaller, indicating a decreasing sequence.
• If the common difference is zero, all terms in the sequence will be identical.

Recognizing the common difference helps you identify how the sequence changes from one term to the next. It is the backbone of arithmetic sequences, keeping them predictable and structured.

Sequence of Numbers

A sequence of numbers is a set of numbers in a specific order. In arithmetic sequences, each number is obtained by adding the common difference to the previous term. Let's talk about constructing the sequence using our given problem:

• First term \(a_1\): Start with the value given for the first term, in this case, \(0\).
• Subsequent terms: Add \(\frac{2}{3}\) to each preceding term to get the next one.
• Example sequence: \(0, \frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}\).

This predictable pattern of numbers is the essence of arithmetic sequences, and they can easily be extended further by continuing to add the common difference to the last known term.

Nth Term Formula

The nth term formula is the key to calculating any term in an arithmetic sequence without listing all previous terms. It's expressed as:
\[ a_n = a_1 + (n - 1) \cdot d \]This formula is intuitive once you get the hang of it:

• \(a_n\): the term you're calculating.
• \(a_1\): the first term of the sequence, a given value to start the calculations.
• \(n-1\): represents the position of the term minus one, a simplification to calculate the number of differences to apply.
• \(d\): the common difference, it completes the multiplication, setting how much you add to the first term \(a_1\).

Whenever you need to find a particular term in the sequence, just substitute \(n\), \(a_1\), and \(d\) into the formula. This process underlines the simplicity and beauty of arithmetic sequences, allowing quick calculations of terms at any desired position.