Problem 31 - Algebra and Trigonometry

Question

Find the indicated term of each geometric sequence. 8th term of \(0.4,0.04,0.004, \ldots\)

Step-by-Step Solution

Answer

The 8th term is 0.00000004.

Step 1: Identify the first term (a) of the sequence

The first term of the geometric sequence is given as 0.4.

Step 2: Determine the common ratio (r)

To find the common ratio, divide the second term by the first term: \[ r = \frac{0.04}{0.4} = 0.1 \]

Step 3: Use the formula for the nth term of a geometric sequence

The formula for the nth term \( a_n \) of a geometric sequence is given by \( a_n = a \cdot r^{(n-1)} \). Here, \( a = 0.4 \), \( r = 0.1 \), and \( n = 8 \).

Step 4: Substitute the values into the formula

Substitute \( a = 0.4 \), \( r = 0.1 \), and \( n = 8 \) into the formula \( a_n = a \cdot r^{(n-1)} \): \[ a_8 = 0.4 \cdot (0.1)^{(8-1)} \]

Step 5: Calculate the exponent

Calculate \( (0.1)^{(8-1)} \): \[ (0.1)^7 = 10^{-7} = 0.0000001 \]

Step 6: Complete the multiplication

Now multiply \( 0.4 \cdot 0.0000001 \): \[ a_8 = 0.4 \cdot 0.0000001 = 0.00000004 \]

Key Concepts

First Term

In a geometric sequence, the first term is the starting point of the sequence. It is usually denoted by the symbol \(a\). For instance, in the given sequence \(0.4, 0.04, 0.004, \ldots\), the first term is \(0.4\). This term sets the stage for the entire sequence and is essential in determining the subsequent terms.
By identifying the first term, you begin to see how the sequence will unfold, enabling you to explore other concepts like the common ratio and the nth term.

Common Ratio

The common ratio, denoted by \(r\), is a key component in a geometric sequence. It is the factor by which you multiply each term to get the next term. To find the common ratio, divide the second term by the first term:

\[ r = \frac{0.04}{0.4} = 0.1 \]
Knowing the common ratio is crucial because it helps determine the pattern of the sequence. In this example, every term is \0.1\ times the previous term. By understanding the common ratio, you can predict any term in the sequence if you know its position.

Nth Term Formula

The nth term formula of a geometric sequence allows you to find any term in the sequence without listing all the previous terms. The formula is given by:

\[ a_n = a \cdot r^{(n-1)} \]
Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term you want to find.
By substituting the values from our problem into the formula, we get:

\[ a_8 = 0.4 \cdot (0.1)^{(8-1)} \]
This formula is extremely powerful because it eliminates the need for repetitive multiplication and directly provides the term you're looking for.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. In the context of our nth term formula, it shows up as \(r^{(n-1)}\). For example, in finding the 8th term of our geometric sequence, we calculate:

\[ (0.1)^{(8-1)} = (0.1)^7 = 10^{-7} = 0.0000001 \]
Exponentiation simplifies the process of repeatedly multiplying the common ratio. It's particularly useful in geometric sequences where terms grow (or shrink) exponentially. Therefore, exponentiation helps us understand the rapid rate of change within these sequences.