Problem 14 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=0.0237, r=10 $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = 0.0237 \cdot 10^{n-1}\). The first five terms are 0.0237, 0.237, 2.37, 23.7, and 237.

Step 1: Identify the given values

Here are the given values: The first term of the sequence, \(a_1\), is 0.0237 and the common ratio, r, is 10.

Step 2: Write down the explicit formula

Using the formula \(a_n = a_1 \cdot r^{n-1}\), we substitute the given values into the formula to get the explicit formula for this sequence: \(a_n = 0.0237 \cdot 10^{n-1}\).

Step 3: Generate the first five terms

Use the explicit formula to generate the first five terms:1. \(a_1= 0.0237 \cdot 10^{1-1} = 0.0237\)2. \(a_2= 0.0237 \cdot 10^{2-1} = 0.237\)3. \(a_3= 0.0237 \cdot 10^{3-1} = 2.37\)4. \(a_4= 0.0237 \cdot 10^{4-1} = 23.7\)5. \(a_5= 0.0237 \cdot 10^{5-1} = 237\)

Key Concepts

Explicit Formula

An explicit formula is used to determine any term in a sequence directly, without needing to compute all the prior terms first. This type of formula provides a clear mathematical expression that relates the position of a term to its value. In the context of geometric sequences, the explicit formula is expressed as:

• \(a_n = a_1 \cdot r^{n-1}\)

This formula calculates the nth term \(a_n\) of the sequence by multiplying the first term \(a_1\) with the common ratio \(r\) raised to the power of \(n-1\). Each variable here has a pivotal role:

• \(a_n\): The term at the nth position.
• \(a_1\): The initial, or first term, of the sequence.
• \(r\): The common ratio, which determines how each term relates to the next.
• \(n\): The position of the term in the sequence.

Understanding this formula is crucial for generating terms in a geometric sequence efficiently.

Common Ratio

The common ratio in a geometric sequence is a fundamental element. It's a constant factor by which each term of the sequence is multiplied to get the next term. Essentially, if you have a sequence, the common ratio \(r\) is determined by dividing any term by the previous term (as long as the previous term is not zero).

• For example, if you observe a sequence like 2, 4, 8, 16, ... the common ratio \(r\) is 2.
• This is because 4 divided by 2 equals 2; similarly, 8 divided by 4 equals 2, and the pattern continues.

In the provided exercise, the common ratio is given as 10, indicating a rapid increase in the sequence elements. This means every term is obtained by multiplying the previous term by 10, illustrating how quickly a geometric sequence can grow with a larger ratio.

Understanding the common ratio helps tremendously in predicting the behavior of the sequence and efficiently calculating future terms.

First Term

The first term of a geometric sequence, denoted as \(a_1\), is the starting point of the sequence and is vital in defining the sequence completely. It is the initial value from which all other terms are derived in conjunction with the common ratio. The equation \(a_n = a_1 \cdot r^{n-1}\) highlights the first term as the base component that each subsequent term builds upon.

• For example, if \(a_1 = 0.0237\), the pattern begins with this value.
• All following terms are calculated by progressively multiplying by the common ratio.

This base value directly affects all future values of the sequence. Identifying \(a_1\) correctly is crucial because a mistake here will cascade down, affecting all further calculations and understanding of the sequence.

Sequence Generation

Sequence generation refers to the process of determining the individual terms of a sequence through a mathematical rule or formula. Using the explicit formula for a geometric sequence, terms can be generated one-by-one or several at a time. Here's how it works:

• Begin with the first term \(a_1\).
• Use the explicit formula \(a_n = a_1 \cdot r^{n-1}\) repeatedly for each term you wish to calculate.
• Proceed sequentially or jump to any term using its index directly bypassing earlier calculations, which would take time otherwise.

In the original exercise, this process is illustrated by using the formula to compute the first five terms:

• \(a_1 = 0.0237\)
• \(a_2 = 0.237\)
• \(a_3 = 2.37\)
• \(a_4 = 23.7\)
• \(a_5 = 237\)

This approach ensures you can generate any term individually or in groups without manual recurrence from terms before it, leveraging the power of the formula and the detailed understanding of the sequence's initial parameters.