Problem 7 - Algebra and Trigonometry

Question

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=\frac{5}{2}, d=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The 10th term is -2.

Step 1: Understanding the Arithmetic Sequence Formula

The formula to find the nth term of an arithmetic sequence is given by: \( a_n = a + (n-1) imes d \), where \( a \) is the first term and \( d \) is the common difference. We will use this formula to find the 10th term, meaning \( n = 10 \).

Step 2: Substituting the Values into the Formula

Substitute the values of \( a = \frac{5}{2} \), \( d = -\frac{1}{2} \), and \( n = 10 \) into the formula: \( a_{10} = \frac{5}{2} + (10-1) \times (-\frac{1}{2}) \).

Step 3: Calculating the Expression Inside the Bracket

Calculate \((10-1) \times (-\frac{1}{2})\). First compute \(10 - 1 = 9\) and then multiply by \(-\frac{1}{2}\): \(9 \times -\frac{1}{2} = -\frac{9}{2} \).

Step 4: Finding the 10th Term

Now, plug \(-\frac{9}{2}\) back into the formula: \( a_{10} = \frac{5}{2} - \frac{9}{2} \). Simplify it to \( a_{10} = \frac{5-9}{2} = -\frac{4}{2} = -2 \).

Key Concepts

nth term formula

The nth term formula is a crucial tool when working with arithmetic sequences. It allows you to find any term in the series without listing all the terms. This formula is: \[ a_n = a + (n-1) \times d \]

• \(a_n\) is the nth term we want to find.
• \(a\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the term number you are looking for.

To use this formula, just follow these simple steps:1. Identify your first term \(a\) and common difference \(d\).2. Decide which term \(n\) you want.3. Substitute \(a\), \(d\), and \(n\) into the formula.
In the given exercise, this formula was used to discover the 10th term of the sequence using \(a = \frac{5}{2}\), \(d = -\frac{1}{2}\), and \(n = 10\). By substituting these values into the formula, you can easily calculate or even predict any term in the sequence, reinforcing that understanding initial values is key to using the formula successfully.

common difference

The common difference \(d\) is a fundamental characteristic of any arithmetic sequence. It defines the amount you add or subtract from one term to the next.

• If \(d\) is positive, the sequence increases.
• If \(d\) is negative, the sequence decreases.
• If \(d\) is zero, all terms are the same.

In our example, the common difference \(d\) is \(-\frac{1}{2}\), which means each subsequent term in the sequence is \(\frac{1}{2}\) less than the previous term. Understanding the common difference helps predict how the sequence behaves as you progress through its terms.
By realizing whether your sequence grows or shrinks, you can make strategic adjustments in calculations or when simply analyzing patterns.

first term

The first term \(a\) is the initial value from which the arithmetic sequence begins. In any arithmetic sequence, the first term sets the starting point for all future calculations and serves as the base to which the common difference is repeatedly added. In our example, the first term \(a\) is given as \(\frac{5}{2}\). This particular value tells you the starting number of the sequence and heavily influences the subsequent terms when combined with the common difference.

• Knowing the first term is essential for utilizing the nth term formula.
• Any mistake in identifying \(a\) would lead to inaccuracies throughout the sequence.

Remember, having a strong grasp of the first term is just as important as understanding the common difference and the formula itself, ensuring you can effectively apply them to a variety of problems and challenges in arithmetic sequences.