Problem 17 - Algebra 2

Question

Find the 32nd term of each sequence. \(3,1,-1,-3, \dots\)

Step-by-Step Solution

Answer

-59

Step 1: Identify the first term and the common difference

Looking at the sequence, the first term \(a_1\) is 3 and the common difference d (the amount subtracted to go from one term to the next) is -2. In this sequence, each term decreases by 2 compared to the previous term.

Step 2: Apply the arithmetic sequence formula

To find the 32nd term, apply the formula for the nth term of an arithmetic sequence, which is \(a_n = a_1 + (n-1)d\). So the 32nd term \(a_{32}\) is given by \(a_{32} = a_1 + (32-1)d = 3 + 31*(-2)\).

Step 3: Calculate the 32nd term

Extend the calculation to find the 32nd term. This comes out as \(a_{32} = 3 - 62 = -59\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key concept that defines the sequence. It is the consistent amount added or subtracted from one term to the next in the sequence. Understanding this concept is essential for identifying and working with arithmetic sequences.

In the exercise sequence "3, 1, -1, -3, ", the common difference is -2. This means that each term decreases by 2 to get to the next term:

• From 3 to 1, subtract 2.
• From 1 to -1, subtract 2.
• From -1 to -3, subtract 2.

This pattern of subtracting 2 continues throughout the sequence. Recognizing the common difference is crucial in using formulas to find specific terms within the sequence.

nth Term Formula

The nth term formula of an arithmetic sequence allows us to determine any term in the sequence without listing all preceding terms. It is essential when dealing with large sequences or when finding terms far below or above the starting point.

The formula is expressed as:\[ a_n = a_1 + (n-1) \times d \]where:

• \(a_n\) is the nth term you are looking for.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number.
• \(d\) is the common difference.

In our example, to find the 32nd term, we use:\[ a_{32} = 3 + (32-1) \times (-2) \]Leading to:\[ a_{32} = 3 - 62 = -59 \]

Sequence Terms

Sequence terms are the individual elements or numbers that make up a sequence in arithmetic sequences. Understanding how these terms are calculated and relate to one another using a formula helps in unlocking the pattern within the sequence.

In our exercise, the sequence begins with 3. Each following term is determined by the common difference, a consistent arithmetic step from one term to the next:

• The first term \(a_1 = 3\),
• The second term \(a_2 = 3 + (-2) = 1\),
• The third term \(a_3 = 1 + (-2) = -1\),
• The fourth term \(a_4 = -1 + (-2) = -3\).

This regular pattern forms the structure of the sequence. Knowing this sequence structure helps you predict future terms easily using the formula.