Problem 15 - 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, a1, and common ratio, r. Find \(a_{8}\) when \(a_{1}=1,000,000, r=0.1.\)

Step-by-Step Solution

Answer

The 8th term of the geometric sequence is 10.

Step 1: Understanding the Problem

We are given a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. We are required to find the 8th term \(a_{8}\) of the sequence, given that the first term \(a_{1} = 1,000,000\), and the common ratio \(r = 0.1\). We will be using the general formula for the n-th term of a geometric sequence, which is \(a_n = a_1 \cdot r^{(n-1)}\).

Step 2: Applying the Geometric Sequence Formula

Let's plug \(a_{1} = 1,000,000\), \(r = 0.1\), and \(n = 8\) into the formula \(a_n = a_1 \cdot r^{(n-1)}\). So, \(a_{8} = 1,000,000 \cdot (0.1)^{8-1} = 1,000,000 \cdot (0.1)^{7}\).

Step 3: Calculating the 8th Term

Now we compute \(a_{8}\): \(a_{8} = 1,000,000 \cdot (0.1)^{7}\). This calculation will give us the 8th term of the sequence.

Key Concepts

nth term

The nth term in a geometric sequence refers to any term in the sequence whose position, "n", you wish to determine. A geometric sequence is a series of numbers where each term after the first is calculated by multiplying the previous term by a constant known as the common ratio. The formula to determine the nth term, sometimes called the general term, is \( a_n = a_1 \cdot r^{(n-1)} \). This formula serves as a guide to reach any specific term in the sequence using the first term \(a_1\), the common ratio \(r\), and the position \(n\).
For instance, if you are tasked with finding the 8th term \(a_8\) in a sequence where the first term \(a_1\) is 1,000,000 and the common ratio \(r\) is 0.1, you will substitute these values into the formula:

• Start with the formula: \( a_n = a_1 \cdot r^{(n-1)} \)
• Substitute the values: \( a_8 = 1,000,000 \cdot (0.1)^{8-1} \)
• Simplify the exponent: \( (0.1)^7 \)

By solving this, you can efficiently identify any term's value in the sequence without compiling all preceding terms.

common ratio

The common ratio is a fundamental component of a geometric sequence. It describes the factor by which you multiply each term to obtain the next term. Denoted by "r", the common ratio is consistent throughout the sequence.
A geometric sequence can be understood by looking at its first term followed by its progression. Consider a sequence starting at 500, with a common ratio of 0.5. This implies that each succeeding term is half of the previous term:

• First term \(a_1 = 500\)
• Second term \(a_2 = 500 \times 0.5 = 250\)
• Third term \(a_3 = 250 \times 0.5 = 125\)

Recognizing the common ratio is essential for calculating any specific term in the sequence or when verifying that a sequence is geometric. It affects the nature of the sequence significantly, determining whether the sequence values increase, decrease, remain constant, or oscillate.
In the provided exercise, the common ratio is 0.1, demonstrating a rapid decrease in term value, showcasing the dramatic impact a small common ratio can have over several terms.

geometric formula

The geometric formula \( a_n = a_1 \cdot r^{(n-1)} \) is key to identifying specific terms in a geometric sequence. This formula combines the starting point of the sequence and its constant multiplying factor, the common ratio, to unveil any term's value.
Breaking down the components:

• \(a_n\): The term you want to find.
• \(a_1\): The first term in the sequence.
• \(r\): The common ratio, indicating how much to multiply each term by to reach the next term.
• \(n\): The position of the term in the sequence.

This formula helps in quickly calculating an advanced term without iterating through the entire sequence.
For example, to find the 8th term in a sequence where the initial term is 1,000,000 and the common ratio is 0.1, you apply:

• \(a_8 = 1,000,000 \cdot (0.1)^{8-1} \)

Following this formula, you can derive the 8th term directly, showcasing the formula's power and efficiency in managing geometric sequences.