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, a1, and common ratio, r. Find \(a_{\mathrm{s}}\) when \(a_{1}=40,000, r=0.1.\)

Step-by-Step Solution

Answer

The fifth term of the given geometric sequence is 4.

Step 1: Identify Given Terms

From the problem, we identify that the first term of the geometric sequence \(a_1\) is 40,000 and the common ratio \(r\) is 0.1. The term to find is \(a_5\). This means \(n=5\).

Step 2: Apply the nth term formula

We apply the formula for the nth term of a geometric sequence, which is \(a_n = a_1 \cdot r^{(n-1)}\). Substituting the given values into the formula, we get \(a_5 = 40000 \cdot (0.1)^{(5-1)}\).

Step 3: Simplify the expression

Now following the order of operations (Parentheses, Exponents, Multiplication and Division, lastly Addition and Subtraction), we first calculate the exponent part. So,we evaluate \(0.1^{(5-1)}= 0.1^4=0.0001\). Then we multiply this number by 40000. The result is \(a_5 = 40000 \cdot 0.0001 = 4\).

Key Concepts

nth Term Formula

Understanding the nth term formula is key to mastering geometric sequences. This vital piece of mathematics is used to find any term in a geometric sequence when you know the first term and the common ratio. In the given exercise, the nth term formula is expressed as
\( a_n = a_1 \cdot r^{(n-1)} \)
where \( a_n \) represents the nth term you want to find, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number. To make this concept crystal clear, let's look at the provided example.

• The first term \( a_1 \) is given as 40,000.
• The common ratio \( r \) is 0.1.
• We want to find the 5th term, so \( n = 5 \) here.

By plugging these values into our formula, we get:
\( a_5 = 40000 \cdot (0.1)^{5-1} \)
This formula simplifies the process of finding the specified term of a geometric sequence and eliminates the need to list out all the terms up to the one you are interested in.

Common Ratio

The common ratio in a geometric sequence is the constant factor that each term is multiplied by to get the next term. It's given the symbol \( r \). The power of the common ratio is what gives geometric sequences their distinctive 'multiplying' pattern, unlike arithmetic sequences which have a 'adding' pattern.
In our example, \( r = 0.1 \).

Characteristics of the Common Ratio

• If \( r > 1 \), the sequence will grow larger with each term.
• If \( r = 1 \), all terms in the sequence will be the same as the first term, because multiplying by 1 has no effect.
• If \( 0 < r < 1 \), the terms will get smaller, approaching zero.
• If \( -1 < r < 0 \), the terms will alternate in sign and decrease in absolute value.
• If \( r \leq -1 \), the terms will alternate in sign and grow in absolute value.

Our problem illustrates a decreasing sequence since the common ratio is a fraction less than 1, which results in each term being a tenth of the previous term. This is why \( a_5 \) ends up being significantly smaller than \( a_1 \) despite the large initial value.

Exponential Expressions

Exponential expressions are a cornerstone in understanding geometric sequences because they represent the rapid growth or decay of sequences. They are written in the form \( a \cdot r^n \) where \( a \) represents a constant, \( r \) is the base of the exponential which is the common ratio in our context, and \( n \) is the exponent indicating the term number minus one.
The calculation of \( a_5 \) in our exercise demonstrates an exponential expression: \( 40000 \cdot (0.1)^4 \) is evaluated by first determining the value of \( 0.1^4 \) which is 0.0001, an example of exponential decay.

Demystifying the Exponential Decay

The exponent here tells us how many times we multiply the base by itself. Since the common ratio is less than 1, repeated multiplication leads to a progressively smaller result. This behavior is typical of many natural processes such as radioactive decay, cooling temperatures, and depreciation of assets. In the sequence of our example, each term becomes one-tenth of the previous term, a clear demonstration of how exponential expressions encapsulate the essence of geometric sequences with a common ratio less than one.