Problem 14 - Precalculus: Mathematics for Calculus

Question

Find the \(n\) th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10 th term? $$a=14, \quad d=-\frac{3}{2}$$

Step-by-Step Solution

Answer

The 10th term of the sequence is 0.5.

Step 1: Understand the Arithmetic Sequence Formula

In an arithmetic sequence, the nth term is given by the formula \( a_n = a + (n-1) \cdot d \), where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number we want to find.

Step 2: Identify Known Values

From the problem, we know that the first term \( a = 14 \) and the common difference \( d = -\frac{3}{2} \). We want to find the 10th term, so \( n = 10 \).

Step 3: Plug Values into the Formula

Substitute the known values into the arithmetic sequence formula: \( a_{10} = 14 + (10-1) \cdot \left(-\frac{3}{2}\right) \).

Step 4: Simplify the Expression

Compute \( (10-1) \cdot \left(-\frac{3}{2}\right) \). This simplifies to \(-\frac{27}{2} \) or \(-13.5 \).

Step 5: Calculate the 10th Term

Add the product from the previous step to the first term: \( a_{10} = 14 - 13.5 = 0.5 \). Thus, the 10th term is 0.5.

Key Concepts

nth term formula

The nth term formula is an essential part of understanding arithmetic sequences. It is a mathematical expression that helps us find any term in an arithmetic progression without listing all preceding terms. The formula is given by: \[ a_n = a + (n-1) \cdot d \] Here's what each component represents:

• \( a_n \): The term in the sequence we seek.
• \( a \): The first term of the arithmetic sequence.
• \( d \): The common difference, which remains constant between successive terms.
• \( n \): The term number, indicating the position in the sequence.

To use the formula effectively, you simply plug in the values of \( a \), \( d \), and \( n \) into the equation. This process quickly yields the desired term without unnecessary calculations. For example, in this problem, to find the 10th term, we substituted the values to calculate the term directly.

common difference

The common difference \( d \) in an arithmetic sequence is what makes it unique. It's the fixed amount that you add (or subtract) to get from one term to the next. When you know the common difference, you can predict the pattern of increase or decrease in the sequence. In our example, the common difference \( d \) is \(-\frac{3}{2} \). This means that each term in the sequence is \(1.5 \) units less than the previous one. The common difference can also be a positive number, where each term would increase instead. Identifying \( d \) early is crucial as it guides the placement and calculation of each term. By always adding (or subtracting) \( d \) to/from the previous term, the sequence follows a linear path, making it predictable.

arithmetic progression

Arithmetic progression is a sequential arrangement of numbers characterized by the common difference. All terms share this commonality, resulting in a linear pattern. Every arithmetic sequence is a form of arithmetic progression. The interesting quality of such a sequence is that knowing any term, alongside the first term and the common difference, allows the calculation of any other term using the nth term formula. Here's a quick example:

• If the sequence starts at 14, and the common difference is \(-\frac{3}{2}\), the second term would be \(14 + (-\frac{3}{2}) = 12.5\).
• The third term would be \(12.5 + (-\frac{3}{2}) = 11\), and this pattern continues.

Understanding arithmetic progression is crucial for mastering the related concepts, as it simplifies predicting and generating sequence terms efficiently.