Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{1,3,9,27, \ldots\\}\)

Step-by-Step Solution

Answer

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

Step 1: Identify Initial Term

The first term in the sequence is \(a_1 = 1\). This is the starting point of our sequence, known as the initial term.

Step 2: Identify Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{3}{1} = 3\). This ratio is consistent throughout the sequence.

Step 3: Write the General Formula

The explicit formula for a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). Substitute the initial term and common ratio into the formula: \(a_n = 1 \cdot 3^{n-1}\).

Step 4: Simplify the Formula

Since multiplying by 1 does not change the value, the simplified explicit formula is \(a_n = 3^{n-1}\). This formula allows you to find any term in the sequence.

Key Concepts

Explicit Formula

An explicit formula for a geometric sequence allows us to calculate the value of any term in the sequence without needing to find the previous terms. This provides a direct way to access any term and helps in understanding how the sequence progresses.
To construct an explicit formula, we use the general template:

• \(a_n = a_1 \cdot r^{n-1}\)

where:

• \(a_n\) is the \(n\)th term of the sequence
• \(a_1\) is the initial term
• \(r\) is the common ratio

The key advantage of the explicit formula is efficiency. You only need three pieces of information: the initial term, the common ratio, and the term number you’re interested in, making it very useful in calculations for large values of \(n\). For example, for the sequence in the exercise, you can find any term directly by plugging \(n\) into:

• \(a_n = 1 \cdot 3^{n-1} = 3^{n-1}\)

This eliminates unnecessary computations of earlier terms, saving time and effort.

Common Ratio

In a geometric sequence, the common ratio is the number that each term is multiplied by to get the next term. It is a constant value that dictates how the sequence grows or shrinks. Finding the common ratio is a crucial step in identifying and expressing a geometric sequence accurately.
To find the common ratio, we use the formula:

• \(r = \frac{a_2}{a_1}\)

where \(a_2\) is the second term and \(a_1\) is the first term. For the sequence \(1, 3, 9, 27, \ldots\), the common ratio is computed as:

• \(r = \frac{3}{1} = 3\)

This ratio, \(3\), stays the same throughout the sequence. Each term is just the previous term multiplied by \(r = 3\). Understanding the common ratio is essential for building the explicit formula, as it reveals the pattern followed by the sequence. A positive ratio greater than 1, like 3, means the sequence grows exponentially, while a ratio between 0 and 1 would cause the sequence to decay.

Initial Term

The initial term in a geometric sequence is the first term, often denoted as \(a_1\). This term serves as the stepping stone for generating the rest of the sequence. It is fundamental because every following term depends on it, multiplied by the common ratio.For the sequence \(1, 3, 9, 27, \ldots\), the initial term is straightforward:

• \(a_1 = 1\)

The initial term tells us the starting point of our sequence. Without it, you cannot correctly establish the explicit formula. For the explicit formula \(a_n = 1 \cdot 3^{n-1}\), \(1\) is crucial as it holds the place in computations and ensures each term calculation begins on the right footing, multiplied by growing powers of the common ratio. If we were to change \(a_1\), the entire sequence's behavior would shift, although it would still maintain its geometric nature.