Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{-4,-12,-36,-108, \ldots\\}\)

Step-by-Step Solution

Answer

The explicit formula is \(a_n = -4 \, (3)^{n-1}\).

Step 1: Identify the First Term (\(a_1\))

The first term of the sequence is given directly as the first element in the list. Here, \(a_1 = -4\).

Step 2: Determine the Common Ratio (\(r\))

To find the common ratio, divide the second term by the first term. So, \( r = \frac{-12}{-4} = 3 \). Check with more terms to confirm: \(-36 \div -12 = 3\), so \(r = 3\) is consistent.

Step 3: Write the Explicit Formula

The explicit formula for a geometric sequence is \(a_n = a_1 \, r^{n-1}\). Substituting the values, we get the explicit formula \(a_n = -4 \, (3)^{n-1}\).

Key Concepts

Explicit Formula

The explicit formula of a geometric sequence is a way to describe any term in the sequence using its position in the sequence. It's like a shortcut to find the value of any term without listing all the terms before it. The general form for an explicit formula in a geometric sequence is:

• \(a_n = a_1 \, r^{n-1}\)

Here, \(a_n\) represents the term you want to find. \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. This formula allows us to plug these values in and easily calculate any term's value.
Let’s take a practical example from the sequence \(-4, -12, -36, -108, \ldots\), where you know \(a_1\) is \(-4\) and \(r\) is \(3\). By plugging these into the formula, you quickly get \(a_n = -4 \times (3)^{n-1}\). This means, if you wanted the fourth term, you calculate it directly with \(-4 \times 3^{3}\), achieving \(-108\) without having to find each term one by one.
Understanding the explicit formula is crucial for quick calculations and verifying the correctness of any sequence.

Common Ratio

The common ratio is a fundamental element of any geometric sequence. It is the factor by which we multiply to get from one term to the next. To find it, simply divide any term by its preceding term.

• For example, with the sequence \(-4, -12, -36, -108, \ldots\), find \(r\) by computing \(\frac{-12}{-4}\). Hence, \(r = 3\).

Using another example from the sequence, divide \(-36\) by \(-12\) to verify, and you’ll see it also equals \(3\). This consistency across terms confirms the common ratio.
Why is the common ratio important? Knowing it simplifies finding each term's value using the explicit formula, making computations straightforward. The common ratio distinguishes geometric sequences from other sequence types, like arithmetic sequences, where constant differences exist instead. Recognizing these patterns tells you how the sequence grows or shrinks, which is especially useful for larger sequences.

First Term

Identifying the first term, denoted as \(a_1\), is the initial step in managing any geometric sequence. The first term is simply the sequence's starting point. For our sequence example \(-4, -12, -36, -108, \ldots\), the first term \(a_1\) is \(-4\).
The first term acts as the foundation from which all subsequent terms are derived. It helps calculate further terms using recurrence relations or the explicit formula. Recognizing this crucial number is important because it lays the groundwork for understanding how the sequence unfolds.
In problems where the first term is given, such as the ones involving explicit formulas, make sure to extract and use it accurately. This simple step ensures your calculations are based on solid footing, paving the way to correctly solve more complex problems or sequences.