Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-17,-217,-417, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = -200n + 183 \).

Step 1: Identify the First Term

The first term of the sequence is the initial value provided in the sequence. Here, the first term \( a_1 \) is \(-17\).

Step 2: Determine the Common Difference

To find the common difference \( d \), subtract the first term from the second term: \(-217 - (-17) = -217 + 17 = -200\).

Step 3: Write the General Formula for the Arithmetic Sequence

In an arithmetic sequence, the \( n \)-th term can be expressed as \( a_n = a_1 + (n-1) \times d \). Substitute \( a_1 = -17 \) and \( d = -200 \) into this formula.

Step 4: Simplify the Expression

Substituting the values from Step 3, we have: \( a_n = -17 + (n-1)(-200) \). This simplifies to: \[ a_n = -17 - 200(n - 1) \].Expanding, this becomes: \[ a_n = -17 - 200n + 200 \] which further simplifies to \[ a_n = -200n + 183 \].

Key Concepts

Explicit Formula

An explicit formula in the context of an arithmetic sequence allows us to determine any term in the sequence without needing to know the previous term. In simpler terms, it gives us a quick way to find the value of the \(n\)-th term directly from its position in the sequence.
The explicit formula for an arithmetic sequence is given by:

• \(a_n = a_1 + (n-1) \times d \)
• \(a_1\) is the first term of the sequence.
• \(d\) represents the common difference, or how much we add or subtract to get from one term to the next.

This formula helps us understand the pattern within the sequence, making it easier to predict subsequent terms.

Common Difference

The common difference is a vital element of an arithmetic sequence. It's the amount you add or subtract to go from one term to the next. Finding the common difference is your first step towards understanding the structure of the sequence.
To calculate the common difference, you subtract the first term from the second term. In our example:

• Second term: -217
• First term: -17
• Common difference \(d = -217 - (-17) = -200\)

This value is consistent throughout the sequence, underscoring the uniform nature of arithmetic progressions. Knowing this helps us predict future terms and construct the explicit formula.

n-th Term

The \(n\)-th term in an arithmetic sequence refers to a specific term where \(n\) indicates its position. When we talk about the \(n\)-th term, we mean the formula that represents any term based on its numeric position within the sequence.
Using the explicit formula:

• \(a_n = a_1 + (n-1) \times d\)
• where \(a_n\) is the \(n\)-th term
• \(n\) is the position of the term

The beauty of this formula is that it allows you to find any term in the sequence without having to list out all prior terms. For instance, you can find the 50th or 100th term directly.

Sequence

A sequence is essentially an ordered list of numbers. In the case of arithmetic sequences, this list is structured such that the difference between consecutive numbers remains constant.
An arithmetic sequence looks something like this:

• \(-17, -217, -417, \dots \)

The constant difference defines what type of sequence it is. In this sequence, each term is decreasing by 200, which is evident from the common difference. Sequences like these are crucial in understanding patterns and progressions in mathematics and other related fields.