Problem 16 - College Algebra

Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) Find \(a_{8}\) when \(a_{1}=40,000, r=0.1\)

Step-by-Step Solution

Answer

Therefore, the 8th term of the geometric sequence is 4.

Step 1: Identify given variables

First, let's identify the given variables in the problem: \(a_{1} = 40000\), \(r = 0.1\) and \(n = 8\).

Step 2: Apply the formula

Now, we apply the formula for finding the nth term of a geometric sequence, which is \(a_n = a_1 * r^{(n-1)}\). Here, \(a_8 = 40000 * (0.1)^{7}\).

Step 3: Compute the value

After substituting the values into the formula, compute the value of \(a_8\). The expression becomes: \(a_8 = 40000 * (0.1)^{7} = 4\)

Key Concepts

nth Term of a Geometric Sequence

Understanding the nth term of a geometric sequence is essential when studying sequences and series. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Let's take a deeper look. If you know the first term of a geometric sequence, which is often denoted as \( a_1 \), and you want to find any term in the sequence (the nth term), the formula \( a_n = a_1 \times r^{(n-1)} \) comes into play. In this formula, \( a_n \) represents the nth term you’re looking for, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

For example, consider a situation where you're given the first term of a sequence, say \( a_1 = 40,000 \), a common ratio of \( r = 0.1 \), and you want to find the 8th term (\( a_8 \)). According to the formula, you’d calculate it as follows: \( a_8 = 40,000 \times (0.1)^{8-1} = 40,000 \times (0.1)^7 = 4 \). The 8th term in this sequence is 4.

Common Ratio

The common ratio in a geometric sequence is arguably the heartbeat of the sequence, determining its growth or decay. It is represented by the variable \( r \) and is constant for any successive terms in a geometric sequence. This ratio can be computed by dividing any term in the sequence by the preceding term.

For instance, if you take a sequence where the second term (\( a_2 \)) is 10 and the first term (\( a_1 \)) is 2, the common ratio would be \( r = \frac{a_2}{a_1} = \frac{10}{2} = 5 \). This ratio shows how much each term is multiplied by to get the next term. In the previous exercise involving the 8th term of a geometric sequence, the common ratio \( r \) was 0.1, indicating that each term is a tenth of the previous term, which signifies an exponential decay.

Sequence and Series

When we talk about sequences and series, we're venturing into an area of mathematics that deals with ordered lists of numbers and their summation. A sequence is a set of numbers in a specific order, while a series is the sum of the terms of a sequence.

For geometric sequences, the terms can either grow rapidly or decrease rapidly, depending on whether the common ratio is greater than one or between zero and one, respectively. When adding the terms of a geometric sequence together, you get a geometric series.

Convergent and Divergent Series

It's also important to distinguish between convergent and divergent series. A geometric series is convergent if the common ratio’s absolute value is less than one, meaning it has a sum. Conversely, if the absolute value of the common ratio is greater than one, the series is divergent, and the sum goes to infinity.

Identifying whether a sequence is arithmetic or geometric and determining its common ratio are foundational skills for working with these mathematical constructs. Being adept at these concepts allows you to understand the properties of the sequence or series at hand and to find other related values, such as the sum of a geometric series, which has its own formula.