Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=x, a_{2}=x+3$$

Step-by-Step Solution

Answer

The eighth term is \(a_8 = x + 21\) and the general term is \(a_n = 3n - 3 + x\).

Step 1: Identify the Common Difference

The given arithmetic sequence starts with \(a_1 = x\) and \(a_2 = x + 3\). To find the common difference \(d\), subtract \(a_1\) from \(a_2\): \(d = (x + 3) - x = 3\).

Step 2: Use the Formula for the General Term

The formula for the \(n\)-th term in an arithmetic sequence is \(a_n = a_1 + (n-1)\cdot d\). Here, \(a_1 = x\) and \(d = 3\). Substitute these values into the formula: \(a_n = x + (n-1)\cdot 3\).

Step 3: Calculate the Eighth Term

To find the eighth term, substitute \(n = 8\) into the formula: \(a_8 = x + (8-1)\cdot 3 = x + 21\). So, \(a_8 = x + 21\).

Step 4: Express the General Term for the Sequence

Based on Step 2, the general term for the sequence is \(a_n = x + 3(n-1)\), which simplifies to \(a_n = 3n - 3 + x\).

Key Concepts

Understanding Common Difference

An arithmetic sequence is a series of numbers with a consistent difference between consecutive terms. This difference is known as the 'common difference'. To identify this, subtract the first term from the second. In our example, the common difference \(d\) is calculated by taking the difference between the second term \(a_2 = x + 3\) and the first term \(a_1 = x\). The resulting calculation, \(d = (x + 3) - x\), simplifies to \(d = 3\). This shows that each term in the sequence increases by 3 compared to the previous term.

• The common difference is what keeps the sequence uniform.
• It's the key to figuring out future terms in the sequence.
• Once determined, solving for any term gets straightforward.

Exploring the n-th Term Formula

The n-th term formula is a powerful tool in any arithmetic sequence. This formula provides a way to find any term in the sequence, given its position \(n\). The general form of the n-th term formula in an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\).

First, identify your initial term, \(a_1\), and your common difference, \(d\). For our sequence, \(a_1 = x\) and \(d = 3\). Substituting these into the formula, we get \(a_n = x + (n-1) \cdot 3\), providing a blueprint to compute any term's value in the sequence. It's important as it enables direct calculation of terms without having to write out the entire sequence.

Determining the General Term

The general term of an arithmetic sequence describes any term's position in terms of a formula. The general term formula for our exercise, based on the given information and n-th term formula, becomes \(a_n = 3n - 3 + x\). This form represents every term in the sequence by expressing a relationship between the position \(n\) and the corresponding term value.

Key points to remember:

• The general term is crucial for expressing any term in the sequence compactly.
• It allows you to find specific terms quickly, such as the 8th term, without excessive calculations.
• This efficiency is important for working with large sequences or when solving problems that require term identification.

Knowing the general term offers a deeper understanding of the sequence's structure and behavior, simplifying analysis and problem-solving.