Problem 78 - Algebra 2

Question

In an arithmetic sequence with \(a_{1}=2\) and \(d=-2,\) which term is \(-82 ?\)

Step-by-Step Solution

Answer

The -82 is the 43rd term in the arithmetic sequence.

Step 1: Understanding the Problem

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference. For any arithmetic sequence, the nth term \(a_{n}\) can be calculated using the formula: \(a_{n}=a_{1}+d(n-1)\), where \(a_{1}\) is the first term, d is the common difference, and n is the term number.

Step 2: Substitute the Given Values into the Formula

In this exercise, we're given \(a_{1}=2, d=-2, a_{n}=-82 \). We can substitute these values into our formula: -82 = 2 - 2(n - 1).

Step 3: Solve the Equation for n

To solve for n, first simplify the equation: -82 = 2 - 2n + 2. Combine like terms to get: -82 = 4 - 2n. Next, subtract 4 from both sides of the equation: -86 = -2n. Finally, divide each side by -2: n = 43.

Key Concepts

Exploring the Common Difference

An arithmetic sequence is defined by its consistent pattern of differences between consecutive terms. This consistent value is known as the "common difference," represented by the letter \(d\). For instance, in the sequence provided, the common difference \(d\) is \(-2\).

This tells us that each subsequent term in the sequence is found by subtracting 2 from the previous term. The concept of the common difference is crucial as it dictates the overall pattern of the sequence. Essentially, it is the backbone of any arithmetic sequence, ensuring that the numbers are generated in a predictable manner.

• A positive common difference will increase the value of each term.
• A negative common difference will decrease the value of each term.
• When the common difference is zero, the sequence becomes constant.

Understanding and identifying the common difference is the first step in solving problems related to arithmetic sequences. This is because it helps us know how each term relates to its neighbors in the sequence. Recognizing and calculating \(d\) correctly allow us to predict any term in the sequence using simple arithmetic operations.

Unraveling the Nth Term Formula

The "nth term formula" is the magic tool that helps us find any term in an arithmetic sequence without listing all the terms. The formula is given by:

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

Here, \(a_{n}\) is the term we're looking to find, \(a_{1}\) is the first term, \(d\) is the common difference, and \(n\) is the term's position.The nth term formula allows for quick calculation and identification of terms in any arithmetic sequence. Let's walk through its application:

• Identify the first term \(a_{1}\) and the common difference \(d\).
• Substitute these values and the desired position \(n\) into the formula.
• Solve the equation to find the term \(a_{n}\) or determine \(n\) if the term and other variables are given.

In our exercise, we used this formula to substitute the given values and solve for the position \(n\) when the term \(-82\) was given. This powerful formula is key to navigating arithmetic sequences efficiently and effectively.

Decoding Term Identification

Identifying a specific term in an arithmetic sequence involves a bit of detective work and application of both the common difference and the nth term formula. When tasked with finding, say, the position of a term like \(-82\), a methodical approach is vital.

In our exercise, \(-82\) was identified based on its position in the sequence. Here's how:

• Start with the nth term formula: \(a_{n} = a_{1} + d(n - 1)\).
• Substitute \(a_{1}, d,\) and \(a_{n}\) into the formula.
• Solve for \(n\) to find the term's position.

In this case, substituting the values \(a_{1} = 2, d = -2, \text{ and } a_{n} = -82\) into the formula, we calculated that \(n = 43\).

This means that the \(-82\) is the 43rd term in our sequence. This knowledge can intersect with real-world issues where identifying terms correctly in sequences is needed. By practicing how to break down these steps, you build a solid foundation in handling arithmetic sequences.