Problem 6 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{1}=8, r=-5$$

Step-by-Step Solution

Answer

\(a_5 = 5000; \; a_n = 8 \cdot (-5)^{n-1}\)

Step 1: Understanding the Formula

In a geometric sequence, the general term (nth term) is given by the formula: \ a_n = a_1 \cdot r^{(n-1)}, where \(a_1\) is the first term and \(r\) is the common ratio.

Step 2: Find the Fifth Term (a_5)

We need to find the fifth term \(a_5\) using the formula. Substitute \(a_1 = 8\), \(r = -5\), and \(n = 5\) into the formula: \ a_5 = 8 \cdot (-5)^{(5-1)} = 8 \cdot (-5)^4.

Step 3: Calculate a_5

Calculate \((-5)^4\), which is the fourth power of -5. \ (-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 25 \times 25 = 625. \ So, \(a_5 = 8 \cdot 625 = 5000\).

Step 4: General Formula for a_n

Substitute \(a_1 = 8\) and \(r = -5\) into the general formula to express \(a_n\): \ a_n = 8 \cdot (-5)^{(n-1)}.

Key Concepts

nth term formula

In a geometric sequence, the term formula is a key concept. It allows you to find any term in the sequence without having to list all the previous terms. The formula for the nth term is expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where:

• \(a_n\) is the term you want to find,
• \(a_1\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the term position you are looking for.

Generally, substituting the known values into the formula gives you a straightforward way to find any term. Just plug the values into this formula to get your desired term. Remember, understanding this formula helps you to solve many problems concerning geometric sequences.

common ratio

The common ratio is an element that defines how a geometric sequence progresses from one term to the next. It is denoted by \(r\). Here's how you can identify and use it:

• The common ratio is the factor by which each term of the sequence is multiplied to get the next term.
• In the context of our example, \(r = -5\), meaning that each subsequent term is the previous term multiplied by \(-5\).

Having the common ratio not only helps in identifying the pattern of a sequence but also plays a critical role in the nth term formula. A positive common ratio indicates a sequence that increases in magnitude, while a negative ratio suggests alternation between positive and negative terms. Overall, understanding the common ratio is vital for predicting terms and analyzing sequence behaviors.

fifth term calculation

Finding the fifth term \(a_5\) in a geometric sequence involves a simple application of the nth term formula. We already know that \[ a_n = a_1 \cdot r^{(n-1)} \]Substituting the given values for \(a_1 = 8\), \(r = -5\), and \(n = 5\), the equation becomes:\[ a_5 = 8 \cdot (-5)^{4} \]Calculating \((-5)^4\), we multiply \(-5\) by itself four times:

• \((-5) \times (-5) = 25\)
• \(25 \times 25 = 625\)

Thus, \[ a_5 = 8 \cdot 625 = 5000 \]That means the fifth term of the sequence is 5000. This demonstrates the sequence's behavior, alternating due to the negative common ratio and resulting in a positive fifth term here. Recognizing this calculation can help you master finding any specific term in such sequences effectively.