Problem 46 - College Algebra

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 \( a_n = 183 - 200n \).

Step 1: Identify the First Term

The first term of the sequence \( a_1 \) is clearly given as \(-17\). This is the starting point for our arithmetic sequence.

Step 2: Calculate the Common Difference

The common difference \( d \) can be found by subtracting the first term from the second term: \( d = -217 - (-17) = -200 \).

Step 3: Use the Arithmetic Sequence Formula

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \).

Step 4: Substitute Known Values into the Formula

Substitute \( a_1 = -17 \) and \( d = -200 \) into the formula: \( a_n = -17 + (n-1)(-200) \).

Step 5: Simplify the Expression

Simplify the formula to \( a_n = -17 - 200(n-1) \), which can also be expressed as \( a_n = -17 - 200n + 200 = 183 - 200n \).

Key Concepts

Explicit Formula

An explicit formula is a mathematical way to express the general term of an arithmetic sequence. It provides a method to find any term directly, without having to calculate all the preceding terms. This formula can be incredibly helpful, especially when dealing with long sequences.

The explicit formula for an arithmetic sequence looks like this: \[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) represents the nth term in the sequence
• \( a_1 \) is the first term
• \( d \) is the common difference, which is the consistent difference between each term

For example, in the sequence provided in the exercise, the explicit formula allows us to compute any term by simply substituting the values for \( a_1 \), \( n \), and \( d \) directly into the formula. This direct approach saves time and reduces error compared to calculating step-by-step for each term.

Common Difference

The common difference, symbolized as \( d \), is a crucial element in understanding arithmetic sequences. It's the consistent amount that you add (or subtract) to go from one term to the next in the sequence.

To calculate the common difference, subtract any term from the term immediately following it. For instance, if the first two terms of a sequence are known, namely

• \( a_1 = -17 \)
• \( a_2 = -217 \)

the common difference can be found using:\[ d = a_2 - a_1 = -217 + 17 = -200 \]
A negative common difference, as shown in this example, indicates that the sequence decreases by the same amount with each subsequent term.

Understanding the common difference is essential because it not only dictates the direction of the sequence (increasing or decreasing) but also helps in predicting future and past terms in the sequence precisely when used in the explicit formula.

First Term

The first term of an arithmetic sequence, denoted as \( a_1 \), serves as the starting point. It is the reference from which all other terms in the sequence are derived.

Knowing this initial term is vital because each subsequent term is built using the first term along with the common difference. Let's consider our sequence with \( a_1 = -17 \). From here, each following number is calculated by adding the common difference
(-200 in this case) repeatedly.

The importance of the first term becomes more evident when employing the explicit formula, where it serves as a significant part of the calculation. For any given nth term, the first term forms the base value to which a multiple of the common difference is added or subtracted, depending on the sequence's nature.