Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=4 n-7$$

Step-by-Step Solution

Answer

The first five terms of the sequence are -3, 1, 5, 9, 13.

Step 1: Find the First Term

We substitute \(n = 1\) into the sequence formula: \(a_{1} = 4*1 - 7 = -3 \). So, the first term is -3.

Step 2: Find the Second Term

Next, substitute \(n = 2\) into the formula: \(a_{2} = 4*2 - 7 = 1 \). The second term is thus 1.

Step 3: Find the Third Term

We substitute \(n = 3\): \(a_{3} = 4*3 - 7 = 5 \). Thus, the third term of the sequence is 5.

Step 4: Find the Fourth Term

Substituting \(n = 4\) into the formula gives \(a_{4} = 4*4 - 7 = 9 \). Hence, the fourth term of the sequence is 9.

Step 5: Find the Fifth Term

Finally, we substitute \(n = 5\) into the formula: \(a_{5} = 4*5 - 7 = 13 \). The fifth term of the sequence is therefore 13.

Key Concepts

Sequence Formulas

To understand arithmetic sequences, we need to start with sequence formulas. Sequence formulas give us a rule to find any term in the sequence. An arithmetic sequence is a special kind of sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, and it plays a crucial role in the formula.

For any arithmetic sequence, the formula for the general term is given by:

• \[ a_{n} = a_{1} + (n - 1) imes d \]

Here,

• \( a_{n} \) is the term we want to find,
• \( a_{1} \) is the first term of the sequence,
• \( d \) is the common difference, and
• \( n \) is the term number.

For example, in the original exercise given, the sequence formula is \( a_{n} = 4n - 7 \). This formula is already simplified and allows us to find any term in the sequence by just plugging in the value of \( n \).

Term Calculation

Calculating terms in an arithmetic sequence involves substituting different values of \( n \) into the sequence formula. This allows us to calculate each term directly.

Let's walk through the original exercise step by step to illustrate this:

• Find the first term by setting \( n = 1 \): \( a_{1} = 4 \times 1 - 7 = -3 \).
• For the second term, we set \( n = 2 \): \( a_{2} = 4 \times 2 - 7 = 1 \).
• The third term is found by substituting \( n = 3 \): \( a_{3} = 4 \times 3 - 7 = 5 \).
• Substitute \( n = 4 \) to get the fourth term: \( a_{4} = 4 \times 4 - 7 = 9 \).
• Finally, for the fifth term, set \( n = 5 \): \( a_{5} = 4 \times 5 - 7 = 13 \).

This step-by-step calculation helps to see how each term is derived from the formula, showcasing the simple arithmetic operations involved.

Linear Sequences

Arithmetic sequences are also known as linear sequences because they form a straight line when plotted on a graph. The constant difference in arithmetic sequences means that the terms increase or decrease by the same amount each time. This consistency creates a pattern that is easy to predict and represent visually.

To determine if a sequence is linear, check if the difference between each consecutive term is the same.

• For instance, in the sequence from the exercise, the terms are -3, 1, 5, 9, 13.
• The difference from term to term is 4 (e.g., \(1 - (-3) = 4\), \(5 - 1 = 4\), and so on).

This constant difference, known as the common difference, allows us to confirm the linearity of the sequence.

Moreover, when these terms are plotted with terms along the y-axis and their positions along the x-axis, a straight line forms, which is a clear representation of a linear relationship. This characteristic makes them both valuable in mathematics and practical for understanding relationships in various applications.