Problem 72 - Algebra 2

Question

Graph the arithmetic sequence generated by each formula over the domain \(1 \leq n \leq 10 .\) \(a_{1}=-60, a_{n}=a_{n-1}+9\)

Step-by-Step Solution

Answer

The sequence generated by the formulas is -60, -51, -42, -33, -24, -15, -6, 3, 12, 21, and results in a straight line when graphed over the domain 1 <= n <= 10.

Step 1: Determine the Values of the Sequence

Start with the first term, which is given as -60. Then, for each successive term, add 9 to the previous term, up until the 10th term. This produces the sequence -60, -51, -42, -33, -24, -15, -6, 3, 12, 21.

Step 2: Plotting the Sequence

On the x-axis, plot the term numbers (referred to as 'n') from 1 to 10. On the y-axis, plot the corresponding values of the terms calculated in step 1. Start with (1, -60) and end with (10, 21). The points will form a straight line, reflecting the constant difference of the sequence.

Key Concepts

Graphing Sequences

Graphing an arithmetic sequence helps to visualize how the numbers in the sequence change over time. When graphing, each term of the sequence is represented as a point on the graph. In our case, this involves plotting term numbers on the x-axis (from 1 to 10) and their corresponding values on the y-axis.

When plotted, the points of an arithmetic sequence align in a straight line. This linear pattern is due to the constant difference between consecutive terms. For the sequence given, which starts at -60 and increases by 9 each step, the points illustrate a precise upward linear trend. To graph a sequence effectively, follow these tips:

• Label your axes clearly: term numbers (n) on the x-axis and sequence values on the y-axis.
• Plot points sequentially from your first term to your last term.
• Connect the points to see the linear relationship in an arithmetic sequence.

Algebra 2

In Algebra 2, understanding sequences, especially arithmetic ones, is a fundamental skill. It's crucial to recognize how to derive sequences and express them using formulas. An arithmetic sequence or progression is characterized by its constant difference, represented by 'd', which determines how much each term changes as the sequence progresses.

Using algebraic notation and techniques:

• Recognize the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) \, d\).
• Where \(a_1\) is the first term, and \(d\) is the common difference.
• Seamlessly manipulate algebraic expressions to solve and transform sequences.

Algebra 2 concepts aid in establishing a connection between numbers and patterns, providing a clear understanding of sequences in a linear framework.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant difference, known as 'd', to the previous term. This constant addition property is what distinguishes arithmetic progressions from other types of sequences.

The general form of an arithmetic sequence is given by the formula \(a_n = a_1 + (n - 1) \, d\), where:

• \(a_n\) = the nth term
• \(a_1\) = the first term
• \(d\) = the common difference
• \(n\) = the term position in the sequence, which starts from 1

In our example with the sequence \(-60, -51, -42, -33,...\), the first term is -60, and the common difference is 9. Each term is derived by consistently adding 9 to the previous term, forming a smooth sequence. Understanding the framework of arithmetic progressions allows students to predict any term of the sequence and extend the sequence beyond its basic range.