Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{15.8,18.5,21.2, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = 15.8 + (n-1) \cdot 2.7 \).

Step 1: Identify the First Term

The first term of the sequence is given as \( a_1 = 15.8 \). This is the starting point of the arithmetic sequence.

Step 2: Find the Common Difference

To find the common difference \( d \), subtract the first term from the second term. \( d = 18.5 - 15.8 = 2.7 \). This value represents the constant amount added to each term to get the next term.

Step 3: Write the Explicit Formula

An explicit formula for an arithmetic sequence is given by \[ a_n = a_1 + (n-1) \cdot d \]Substitute the values of \( a_1 \) and \( d \):\[ a_n = 15.8 + (n-1) \cdot 2.7 \]

Key Concepts

Explicit Formula

An explicit formula provides a direct way to determine any term in an arithmetic sequence without needing to know the previous terms. This is extremely helpful when you want to find, let's say, the 100th term directly. The explicit formula for an arithmetic sequence is written as:\[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) is the nth term of the sequence
• \( a_1 \) is the first term
• \( d \) is the common difference between terms
• \( n \) is the position of the term in the sequence

The power of the explicit formula lies in its simplicity and efficiency. You can compute any term by substituting the term number into the formula. For example, if you need the 5th term of the sequence \( a_n = 15.8 + (n-1) \cdot 2.7 \), simply plug \( n = 5 \) into the formula and solve. This way, you avoid recalculating every preceding term.

Common Difference

The common difference in an arithmetic sequence is the fixed, constant number you add to a term to get the next term. In our sequence, the common difference \( d \) is found by subtracting the first term from the second term. This gives us:\[ d = 18.5 - 15.8 = 2.7 \]This tells us that each term is 2.7 units larger than the previous term. Here’s how to recognize a common difference:

• It is the same for every pair of consecutive terms.
• It is added repeatedly, leading to evenly spaced numbers.

Knowing the common difference is crucial. It not only tells us the pattern of our sequence but is also a key part of the formula for computing any term. Whenever you have a sequence and you want to describe how it grows or shrinks, start by calculating the common difference.

First Term

The first term of an arithmetic sequence, often denoted as \( a_1 \), is the starting value of the sequence. It is the point from which the arithmetic progression begins. In our case, the first term is:\[ a_1 = 15.8 \]This first term serves as a reference point and is always required to use the explicit formula effectively. Without knowing \( a_1 \), you cannot apply the formula to find subsequent terms.Understanding the first term is very straightforward:

• It is always the initial value in your list of numbers.
• It establishes the baseline for where your sequence starts.

In the context of forming an arithmetic sequence, the first term is akin to the foundation of a building—invaluable and necessary to determine how the rest of the sequence unfolds.