Problem 40 - College Algebra

Question

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\left\\{2, \frac{1}{3}, \frac{1}{18}, \frac{1}{108}, \ldots\right\\} $$

Step-by-Step Solution

Answer

Explicit formula: \( a_n = 2 \left( \frac{1}{6} \right)^{n-1} \).

Step 1: Identify the First Term

The first term of the sequence is given as 2. So, \( a_1 = 2 \).

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{1}{3} \div 2 = \frac{1}{6} \).

Step 3: Write the General Formula

A geometric sequence has an explicit formula of \( a_n = a_1 \, r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio. Substitute the values: \( a_n = 2 \left( \frac{1}{6} \right)^{n-1} \).

Key Concepts

Explicit Formula

In a geometric sequence, the explicit formula is a powerful tool. It allows you to find any term without calculating all the previous ones. The explicit formula for a geometric sequence is given by

• \( a_n = a_1 \, r^{n-1} \),

where \( a_n \) is the term you're trying to find, \( a_1 \) is the first term of the sequence, and \( r \) is the common ratio.
To use this formula, you simply need the first term and the common ratio.
The beauty of the explicit formula is its efficiency. For instance, if you want the 10th term, you plug in \( n = 10 \) along with your values of \( a_1 \) and \( r \). This saves you from having to go through each term sequentially.
In our example, the explicit formula is \( a_n = 2 \left( \frac{1}{6} \right)^{n-1} \).
This sets up a direct way to find any term in the sequence by substituting the desired term number for \( n \).

Common Ratio

The common ratio \( r \) is a defining feature of a geometric sequence. It's the factor by which we multiply to get from one term to the next.
To find the common ratio, you take any term in the sequence, divide it by the previous term, and you'll get \( r \).
In our example, by dividing the second term \( \frac{1}{3} \) by the first term 2, we find that

• \( r = \frac{1}{6} \).

This constant factor is what differentiates geometric sequences from arithmetic ones, where the difference between terms is constant.
The common ratio can be a fraction, as seen here, indicating that the terms decrease in value.
It can also be a positive or negative number, influencing whether the sequence grows or changes direction.

Sequence Term

Each number in a geometric sequence is referred to as a sequence term. Sequence terms follow a predictable pattern set by the explicit formula.
Using the explicit formula, you can find any sequence term easily. Pick the term number, also known as \( n \), substitute it into

• \( a_n = a_1 \, r^{n-1} \)

to calculate that term's value without having to find the entire sequence up to that point.
In our example, if you wanted the fourth term:

• \( a_4 = 2 \left( \frac{1}{6} \right)^{4-1} = 2 \left( \frac{1}{6} \right)^3 \).

Calculate the power and multiply it by the first term to find the sequence term. Through this straightforward calculation, each sequence term becomes accessible.