Problem 29 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$0.3,-0.09,0.027,-0.0081, \ldots$$

Step-by-Step Solution

Answer

The common ratio is -0.3, the fifth term is 0.00243, and the nth term is \(a_n = 0.3 \cdot (-0.3)^{(n-1)}\).

Step 1: Identify the common ratio

To find the common ratio of this geometric sequence, divide the second term by the first term. Thus, \(-0.09 / 0.3 = -0.3\). Therefore, the common ratio \(r\) is \(-0.3\).

Step 2: Calculate the fifth term

Use the formula for any term in a geometric sequence: \(a_n = a_1 \cdot r^{(n-1)}\). Here, \(a_1 = 0.3\), \(r = -0.3\), and \(n = 5\). Calculate:\[a_5 = 0.3 \cdot (-0.3)^{4} = 0.3 \cdot 0.0081 = 0.00243\]

Step 3: Derive the nth term formula for the sequence

The formula for the \(n\) th term, given \(a_1 = 0.3\) and \(r = -0.3\), can be written as:\[a_n = 0.3 \cdot (-0.3)^{(n-1)}\] This formula allows you to calculate the term based on any term number \(n\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor that each term is multiplied by to get the next term. To find it, simply divide any term in the sequence by the previous term.

For the sequence \( 0.3, -0.09, 0.027, -0.0081, \ldots \), find the common ratio \( r \) by dividing the second term \( -0.09 \) by the first term \( 0.3 \):

• \( r = \frac{-0.09}{0.3} = -0.3 \)

This common ratio \( -0.3 \) shows that each term is the previous term multiplied by \(-0.3\). With a common ratio of \(-0.3\), this sequence alternates signs with each subsequent term.

nth Term Formula

The nth term formula is crucial for finding any specific term in a geometric sequence without listing all the terms. The general formula for a term in a geometric sequence is given by:

• \( a_n = a_1 \cdot r^{(n-1)} \)

Where:

• \( a_n \) is the nth term you want to find,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the term number.

For our sequence, \( a_1 = 0.3 \) and \( r = -0.3 \). The nth term can be calculated as:

• \( a_n = 0.3 \cdot (-0.3)^{(n-1)} \)

This formula helps predict the value of any term in the sequence based on its position \( n \). Understanding this formula is essential for exploring patterns and determining specific sequence values.

Fifth Term

Deriving the fifth term in a geometric sequence requires using the nth term formula. Given the formula \( a_n = a_1 \cdot r^{(n-1)} \), you can substitute the known values to find \( a_5 \):

• \( a_5 = 0.3 \cdot (-0.3)^{4} \)

Compute \( (-0.3)^{4} \):

• \( (-0.3)^{4} = 0.0081 \)

Now, plug this back into the formula:

• \( a_5 = 0.3 \cdot 0.0081 = 0.00243 \)

So, the fifth term is \( 0.00243 \). Applying the formula to find the fifth term showcases the power and simplicity of using the nth term formula. It allows for easy calculation without manually multiplying each term in the sequence.