Problem 11 - A Graphical Approach to College Algebra

Question

Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$-4,-12,-36,-108, \dots$$

Step-by-Step Solution

Answer

\( a_5 = -324 \) and \( a_n = -4 \cdot 3^{n-1} \).

Step 1: Identify the Common Ratio

A geometric sequence has a common ratio between consecutive terms. To find it, divide the second term by the first term: \[ r = \frac{-12}{-4} = 3\] The common ratio is \( r = 3 \).

Step 2: Recurrence Formula for Terms

The recurrence formula for a geometric sequence is given by \( a_{n} = a_{1} \cdot r^{n-1} \). Here, \( a_{1} = -4 \) and \( r = 3 \).

Step 3: Calculate \( a_5 \)

Using the formula from Step 2, substitute \( n = 5 \):\[a_{5} = -4 \cdot 3^{5-1} = -4 \cdot 3^{4}\]Calculate \( 3^4 \):\[3^4 = 81\] So, \[a_{5} = -4 \cdot 81 = -324\]

Step 4: General Formula for \( a_n \)

From Step 2, the general formula for any term \( a_n \) is:\[a_{n} = -4 \cdot 3^{n-1}\]This formula allows calculation of any term in the sequence.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a vital concept. It refers to the number you multiply by to get from one term to the next. Here's how it works:

• Take any consecutive terms in the sequence. For the given sequence, these are -4, -12, -36, -108.
• Divide one term by the previous term to find the common ratio. For example, divide -12 by -4.
• Perform the division: \( r = \frac{-12}{-4} = 3 \).

So in this sequence, the common ratio \( r \) is 3. This ratio is consistent for all consecutive terms you divide, indicating you are indeed working with a geometric sequence. Once you know the common ratio, finding other terms becomes straightforward.

Applying the Recurrence Formula

A recurrence formula is a handy tool in geometric sequences. It helps express the pattern of the sequence in algebraic form.For geometric sequences, the recursion is:\[ a_{n} = a_{1} \cdot r^{n-1} \]Here, \( a_{1} \) is the first term of the sequence, which in this case is -4, and \( r \) (our common ratio) is 3. This formula tells us how to calculate any term in the sequence by plugging in the right number for \( n \), which represents the term number.This equation is incredibly useful as:

• It saves time in manually computing each previous term to get to the next one.
• It allows for direct calculation of any desired term of the sequence.

With this knowledge, you can find any term, even those far in the future, simply by knowing the first term and the common ratio.

Mastering Term Calculation

Once you grasp the idea of common ratio and the recurrence formula, calculating specific terms becomes a breeze. Let’s demonstrate how to calculate the 5th term, \( a_5 \):

• Utilize the recurrence formula: \( a_{n} = -4 \cdot 3^{n-1} \).
• Substitute \( n = 5 \) into the formula: \( a_{5} = -4 \cdot 3^{5-1} \).
• Calculate \( 3^4 \), which is 81. Hence, \( a_{5} = -4 \cdot 81 \).
• The result is \( a_{5} = -324 \).

This process illustrates how predictable geometric sequences are. Once you have the formula, it's all about substituting values and performing simple calculations. This method of term calculation applies to finding any term in the sequence, making it a powerful tool in mathematics.