Problem 15 - 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? $$a=\frac{1}{2}, \quad d=-1$$

Step-by-Step Solution

Answer

The 10th term is \(-\frac{17}{2}\).

Step 1: Understanding the Problem

We need to find the 10th term of an arithmetic sequence. The sequence is defined by its first term \( a = \frac{1}{2} \) and common difference \( d = -1 \).

Step 2: General Formula for the n-th term of an Arithmetic Sequence

The formula for the \( n \)-th term \( a_n \) of an arithmetic sequence is given by: \( a_n = a + (n - 1) \times d \).

Step 3: Apply the formula to find the 10th term

We substitute \( n = 10 \), \( a = \frac{1}{2} \), and \( d = -1 \) into the formula: \[ a_{10} = \frac{1}{2} + (10 - 1) \times (-1) \].

Step 4: Calculate (n-1) * d

Calculate \( (10 - 1) \times (-1) = 9 \times (-1) = -9 \).

Step 5: Add first term and the calculated difference to find the 10th term

Now add the result from the previous step to the first term: \[ a_{10} = \frac{1}{2} + (-9) = \frac{1}{2} - 9 = \frac{1}{2} - \frac{18}{2} = -\frac{17}{2} \].

Key Concepts

n-th term

In an arithmetic sequence, each term progresses in a predictable pattern defined by both the first term and the common difference. The "n-th term" refers to any term within this sequence, indexed by the position number \( n \). Understanding the concept of the n-th term is crucial since it allows us to precisely locate any term in the sequence without having to list all the preceding terms.

Consider, for example, a sequence where you want to find the 10th term. Instead of calculating through the previous nine terms, you can directly compute the 10th term using the sequence's defining characteristics: the first term \( a \) and the common difference \( d \). By fully understanding the n-th term in relation to these characteristics, solving related problems becomes substantially easier.

common difference

The common difference in an arithmetic sequence is the consistent interval between consecutive terms. This "difference" is always the same throughout the sequence, allowing any arithmetic sequence to grow linearly.

• This difference is denoted as \( d \), and it dictates how each subsequent term is generated by adding this value to the previous term.
• If \( d \) is positive, the sequence will continuously increase, while a negative \( d \) will cause the sequence to decrease.
• Understanding the common difference is essential for analyzing the nature and behavior of an arithmetic sequence, especially when predicting specific terms.

In the example exercise, the common difference is \( d = -1 \). This means each term is one unit less than the term before it, which explains the decreasing pattern as you move forward in the sequence.

first term

The first term of an arithmetic sequence, denoted by \( a \), is the initial value from which the sequence begins. It serves as the anchor point for developing the entire sequence.

• Knowing the first term is necessary to compute subsequent terms through the arithmetic formula.
• In any sequence, while the common difference defines the progression, the first term sets the starting point.

For the problem at hand, the first term is \( \frac{1}{2} \). This sets the stage for each subsequent term to be developed by continuously adding the common difference (\( d = -1 \)) to this initial value.

formula for arithmetic sequence

The formula for the n-th term in an arithmetic sequence is a powerful tool used to identify any term in the sequence without listing all preceding terms. The formula is given by:\[a_n = a + (n - 1) imes d\]Where:

• \( a_n \) is the n-th term,
• \( a \) is the first term,
• \( d \) is the common difference,
• \( n \) is the term position you are searching for.

This formula allows us to simply plug in our known values to solve for any term. For example, to find the 10th term, you would substitute \( a = \frac{1}{2} \), \( d = -1 \), and \( n = 10 \) into the formula to find \( a_{10} = -\frac{17}{2} \). Using this formula, you can tackle any similar problem with ease by following consistent steps.