Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{-1.25,-5,-20,-80, \ldots\\}\)

Step-by-Step Solution

Answer

The explicit formula is \(a_n = -1.25 \, \cdot \, 4^{n-1}\).

Step 1: Identify Initial Term

The first step is to identify the first term of the sequence, denoted as \(a_1\). In this sequence, the first term \(a_1\) is \(-1.25\).

Step 2: Find Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(-5 \div (-1.25) = 4\). Check by repeating for other terms: \(-20 \div (-5) = 4\) and \(-80 \div (-20) = 4\), confirming \(r = 4\).

Step 3: Write the Explicit Formula

The formula for a geometric sequence is \(a_n = a_1 \, r^{n-1}\). Substitute \(a_1 = -1.25\) and \(r = 4\) into the formula: \[a_n = -1.25 \, \cdot \, 4^{n-1}\].

Key Concepts

Explicit Formula

An explicit formula in the context of a geometric sequence allows us to find any term of the sequence without having to compute all the preceding terms. This formula is crafted in such a way that it directly expresses the nth term of the sequence in relation to its position within the sequence. This is particularly useful for finding terms far into the sequence quickly and efficiently.
For a geometric sequence, the explicit formula is given by: \[a_n = a_1 imes r^{n-1} \] Here, \(a_n\) is the nth term we wish to find, \(a_1\) is the first term of the sequence, and \(r\) is the common ratio.
Using the sequence from the exercise, we identified that the first term \(a_1\) is \(-1.25\) and the common ratio \(r\) is 4. Thus, the explicit formula becomes: \[a_n = -1.25 imes 4^{n-1} \] This formula enables us to plug in any value of \(n\) to quickly determine the corresponding term in the sequence.

Common Ratio

The common ratio is a crucial part of defining a geometric sequence. It is the factor by which we multiply one term to get the next term in the sequence. Identifying the common ratio helps in forming the explicit formula, which in turn allows us to find any term in the sequence.
To find the common ratio, you divide any term in the sequence by the previous term. From our exercise, we calculated the common ratio by dividing the second term \(-5\) by the first term \(-1.25\), resulting in \(4\). To ensure consistency, it's good practice to verify by checking a few more consecutive terms in the sequence.
It's important to note that the common ratio could be any real number. In our sequence, each term is obtained by multiplying the previous term by \(4\). The consistency of this ratio across terms confirms the sequence indeed behaves geometrically.

Geometric Sequence Formula

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the 'common ratio'. The general formula for any geometric sequence combines the first term, the common ratio, and the position of the term within the sequence.
The formula to find the nth term is: \[a_n = a_1 imes r^{n-1} \] Each component of this formula has a specific role:

• \(a_1\) is the first term, setting the starting point of the sequence.
• \(r\) is the common ratio, dictating how each term relates to its predecessor.
• \(n-1\) ensures that the formula adjusts for the position of \(n\) appropriately, starting from \(a_1\).

Using this formula, as seen in the exercise, simplifies calculating any term. Given the parameters \(-1.25\) for \(a_1\) and \(4\) for \(r\), substituting into the geometric sequence formula allows us to explore various terms without progressive calculation, showcasing the elegance and power of the formula.