Problem 6 - Algebra and Trigonometry

Question

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=3(-4)^{n-1} $$

Step-by-Step Solution

Answer

The terms are 3, -12, 48, -192, 768; common ratio \(r\) is -4.

Step 1: Understanding the Formula

The formula for the sequence is given by \( a_{n} = 3(-4)^{n-1} \). This is a geometric sequence with an initial term multiplied by a power of the base number -4. The exponent is \( n-1 \), which starts at 0 for the first term.

Step 2: Calculate the First Term \(a_1\)

For the first term, substitute \( n = 1\) into the formula:\[a_1 = 3(-4)^{1-1} = 3(-4)^0 = 3 \times 1 = 3\]

Step 3: Calculate the Second Term \(a_2\)

Substitute \( n = 2 \) into the formula:\[a_2 = 3(-4)^{2-1} = 3(-4)^1 = 3 \times (-4) = -12\]

Step 4: Calculate the Third Term \(a_3\)

Substitute \( n = 3 \) into the formula:\[a_3 = 3(-4)^{3-1} = 3(-4)^2 = 3 \times 16 = 48\]

Step 5: Calculate the Fourth Term \(a_4\)

Substitute \( n = 4 \) into the formula:\[a_4 = 3(-4)^{4-1} = 3(-4)^3 = 3 \times (-64) = -192\]

Step 6: Calculate the Fifth Term \(a_5\)

Substitute \( n = 5 \) into the formula:\[a_5 = 3(-4)^{5-1} = 3(-4)^4 = 3 \times 256 = 768\]

Step 7: Listing the First Five Terms

The first five terms of the sequence based on our calculations are:1. \(a_1 = 3\)2. \(a_2 = -12\)3. \(a_3 = 48\)4. \(a_4 = -192\)5. \(a_5 = 768\)

Step 8: Determine the Common Ratio \(r\)

The common ratio \( r \) is found by dividing any term by the previous term. Using \( a_2 \) and \( a_1 \): \[r = \frac{a_2}{a_1} = \frac{-12}{3} = -4\]

Step 9: Graphing the Terms

To graph the first five terms, position them on a coordinate system with the term number \( n \) on the x-axis and the value of \( a_n \) on the y-axis. The points will be: \((1, 3), (2, -12), (3, 48), (4, -192), (5, 768)\). Plot and connect these points to visually represent the sequence.

Key Concepts

Common Ratio

The concept of a common ratio is key in understanding geometric sequences. In our sequence, the common ratio \( r \) is the number we multiply each term by to get to the next term. To find \( r \), you can take any term in the sequence and divide it by the preceding term.
For example, looking at the first two terms we calculated:

• The first term \( a_1 = 3 \)
• The second term \( a_2 = -12 \)

To find the common ratio \( r \), you use the formula: \[ r = \frac{a_2}{a_1} = \frac{-12}{3} = -4 \] So, the common ratio \( r = -4 \) for our sequence.
This constant factor \( r \) remains the same for any two consecutive terms, showing the consistent multiplying effect throughout the sequence.

Sequence Terms

In a geometric sequence, each term is derived from the previous one by multiplying by the common ratio. The term formula for our sequence is given by \[ a_{n} = 3(-4)^{n-1} \]. To find the sequence terms, you'd plug in different values of \( n \).
Let's review the process:

• First term \( a_1 = 3 \)
• Second term \( a_2 = -12 \)
• Third term \( a_3 = 48 \)
• Fourth term \( a_4 = -192 \)
• Fifth term \( a_5 = 768 \)

These were calculated by substituting \( n = 1 \), \( n = 2 \), etc., into the formula. It's important to note the alternating pattern of signs due to the negative common ratio \(-4\). Making calculations step-by-step ensures accuracy and understanding of how each term relates to the next.

Graphing Sequences

Graphing a sequence can help visualize how it behaves and changes over the term numbers. To graph the sequence, you plot term number \( n \) on the x-axis and the term value \( a_n \) on the y-axis.
For the sequence terms we calculated, the points are:

• (1, 3)
• (2, -12)
• (3, 48)
• (4, -192)
• (5, 768)

When you plot these points, you will clearly see the rapid alterations due to the common ratio of \(-4\).
The graph will not only illustrate the alternating positive and negative growth but also highlight how quickly these values escalate or decline. This visualization confirms the unique exponential nature of geometric sequences.