Problem 31 - 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

Common ratio \(r = -0.3\); Fifth term: 0.00243; \(n\)-th term: \(a_n = 0.3 \cdot (-0.3)^{n-1}\).

Step 1: Identify the First Term

In a geometric sequence, the first term is typically denoted by \(a_1\). In this sequence, the first term \(a_1\) is 0.3.

Step 2: Determine the Common Ratio

The common ratio \(r\) in a geometric sequence is found by dividing any term by the previous term. Using the first two terms: \(-0.09 \div 0.3 = -0.3\). We can verify this common ratio with other terms: \(0.027 \div (-0.09) = -0.3\) and \(-0.0081 \div 0.027 = -0.3\). Thus, the common ratio \(r = -0.3\).

Step 3: Calculate the Fifth Term

To find the fifth term \(a_5\) of a geometric sequence, we use the formula \(a_n = a_1 \cdot r^{n-1}\). Here, \(n = 5\), so \(a_5 = 0.3 \cdot (-0.3)^{4}\). Calculate \((-0.3)^4 = 0.0081\). Thus, \(a_5 = 0.3 \cdot 0.0081 = 0.00243\).

Step 4: Find the General Formula for the nth Term

The general formula for the \(n\)-th term \(a_n\) in a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\). Substituting the values of \(a_1 = 0.3\) and \(r = -0.3\) gives \(a_n = 0.3 \cdot (-0.3)^{n-1}\).

Key Concepts

Common Ratio

In a geometric sequence, one of the key features to identify is the common ratio.
The common ratio is the factor by which we multiply one term to get the next term. It remains constant throughout the sequence. To determine the common ratio, analyze the given sequence. Typically, you divide any term by the term before it. For example, in the sequence:

• First term: 0.3
• Second term: -0.09

Divide the second term by the first term: \[-0.09 \div 0.3 = -0.3\]Repeating this process with subsequent terms, like \(0.027 \div (-0.09)\) and \(-0.0081 \div 0.027\), we confirm that the common ratio is indeed \(-0.3\). This process is crucial for ensuring that the sequence is geometric and for understanding how the sequence progresses.

nth Term Formula

The nth term formula of a geometric sequence helps us find any term in the sequence without listing all the terms beforehand. This is especially useful for large sequences. The formula is expressed as: \[a_n = a_1 \cdot r^{n-1}\]Here,

• \(a_n\) denotes the nth term
• \(a_1\) is the first term of the sequence
• \(r\) is the common ratio
• \(n\) is the term number

For example, to find the fifth term in our sequence starting with 0.3 and having a common ratio of -0.3, substitute as follows: \[a_5 = 0.3 \cdot (-0.3)^{5-1} = 0.3 \cdot (-0.3)^4 = 0.3 \cdot 0.0081 = 0.00243\]Using this formula, you can compute any term directly by substituting the appropriate values for \(n\), making it a powerful tool for understanding geometric sequences.

Geometric Series

A geometric series is the sum of the terms in a geometric sequence. Understanding how to manipulate these series can be useful for both mathematical calculations and real-world applications, such as financial calculations involving compound interest. In a geometric sequence, the terms grow (or shrink) exponentially according to the common ratio. The sum is expressed as:\[S_n = a_1 \left( \frac{1 - r^n}{1 - r} \right)\] for \( |r| < 1 \), and \[S_n = a_1 (r^n - 1) / (r - 1)\] for \( r > 1 \).

• \(S_n\) is the sum of the first \(n\) terms,
• \(a_1\) is the first term,
• \(r\) is the common ratio.

These formulas help you understand the total value of all the elements up to a particular term. Exploring geometric series offers insights into the accumulation process when you continuously multiply by a constant factor. This understanding is beneficial in scenarios where growth accelerates over time.