Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ x-8, x-3, x+2, x+7, \ldots $$

Step-by-Step Solution

Answer

\(n\)th term: \(5n + x - 13\), fifth term: \(x + 12\), tenth term: \(x + 37\).

Step 1: Identify the first term and common difference

In an arithmetic sequence, the common difference (\(d\)) is constant between consecutive terms. Here, the first term \(a_1\) is \(x-8\), and the second term \(a_2\) is \(x-3\). To find \(d\), subtract the first term from the second: \(d = (x-3) - (x-8) = 5\).

Step 2: Write the formula for the \(n\)th term

The general formula for the \(n\)th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). Use the first term \(a_1 = x-8\) and the common difference \(d = 5\) to write: \(a_n = (x-8) + (n-1) \cdot 5\).

Step 3: Simplify the \(n\)th term formula

Simplify the expression to find a cleaner form for the \(n\)th term. \[ a_n = (x-8) + 5(n-1) = x - 8 + 5n - 5 = 5n + x - 13 \]. Thus, \(a_n = 5n + x - 13\).

Step 4: Find the fifth term (\(n=5\))

Substitute \(n=5\) into the formula for the \(n\)th term: \(a_5 = 5(5) + x - 13 = 25 + x - 13 = x + 12\).

Step 5: Find the tenth term (\(n=10\))

Substitute \(n=10\) into the formula for the \(n\)th term: \(a_{10} = 5(10) + x - 13 = 50 + x - 13 = x + 37\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a critical element. It represents the consistent increment added from one term to the next. Understanding this difference is crucial because it helps define the behavior of the entire sequence. In the given sequence, the common difference can be found by subtracting the first term from the second. If we have terms like \(x - 8\) and \(x - 3\), the common difference \(d\) is calculated as:

• Subtract the first term from the second: \[d = (x-3) - (x-8) = 5\]

This process tells us that each term in the sequence increases by 5 compared to the previous term. Grasping this concept allows you to determine any missing terms in the sequence quickly. The common difference is always constant in an arithmetic sequence, making it predictable and easy to work with.

Nth Term Formula

The \(n\)th term formula is a useful tool in any arithmetic sequence that helps to identify specific terms without listing all previous ones. The general formula for calculating the \(n\)th term is:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_n\) represents the \(n\)th term in the sequence, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number you are interested in finding. For the given sequence with a first term \(a_1 = x-8\) and a common difference \(d = 5\), the \(n\)th term becomes:

• \(a_n = (x-8) + (n-1) \cdot 5 = 5n + x - 13\)

By utilizing this formula, you can effortlessly plug in any value of \(n\) to find the corresponding sequence term, providing a quick way to expand the sequence further or locate specific terms.

First Term

The first term in an arithmetic sequence sets the foundation for determining all other terms. It is usually represented by \(a_1\) and serves as a starting point from which all other terms are derived. In our example sequence, the first term is given as \(x - 8\). Knowing the first term is essential when working with the \(n\)th term formula because it is the base upon which the entire sequence builds.

• If the first term changes, it affects every term in the sequence.
• The first term combined with the common difference gives the ability to calculate any other term.

For any particular arithmetic sequence task, identifying the first term accurately provides clarity and firm grounding for more complex calculations.

Sequence Terms Calculation

Calculating terms within an arithmetic sequence can be done swiftly using the derived \(n\)th term formula. Once you have established the first term and the common difference, as detailed in previous sections, finding specific terms simply involves substituting the term number into the formula.

• For the fifth term: \(a_5 = 5(5) + x - 13 = x + 12\)
• For the tenth term: \(a_{10} = 5(10) + x - 13 = x + 37\)

Each of these calculations stems from inserting the respective \(n\) value into the formula \(a_n = 5n + x - 13\). This method is efficient, eliminating any need to sequentially calculate each term up to the desired one. It showcases the importance of the \(n\)th term formula as an indispensable tool for navigating through arithmetic sequences and determining terms efficiently.