Problem 22 - Precalculus with Limits

Question

In Exercises 17 - 28, write the first five terms of the geometric sequence \( a_1 = 6, r = -\dfrac{1}{4} \)

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are: 6, -3/2, 3/2, -3/8, 3/16.

Step 1: Calculate the first term

The first term is given, which is \(a_1 = 6\).

Step 2: Calculate the second term

Use the formula with \(i = 2\). So the second term is \(a_2 = a_1 \cdot r^{2-1} = 6 \cdot -\dfrac{1}{4} = -\dfrac{3}{2}\).

Step 3: Calculate the third term

Use the formula with \(i = 3\). So the third term is \(a_3 = a_1 \cdot r^{3-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^2 = \dfrac{3}{2}\).

Step 4: Calculate the fourth term

Use the formula with \(i = 4\). So the fourth term is \(a_4 = a_1 \cdot r^{4-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^3 = -\dfrac{3}{8}\).

Step 5: Calculate the fifth term

Use the formula with \(i = 5\). So the fifth term is \(a_5 = a_1 \cdot r^{5-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^4 = \dfrac{3}{16}\).

Key Concepts

first five terms

To understand a geometric sequence, you need to know how to determine its first few terms. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

In our example, we begin with the first term, which is given as 6. Then, we use the common ratio of \( -\frac{1}{4} \) to find the following terms.

• The second term: Multiply the first term by the common ratio, \( 6 \times -\frac{1}{4} = -\frac{3}{2} \).
• The third term: Continue by multiplying the second term by the common ratio, \( -\frac{3}{2} \times -\frac{1}{4} = \frac{3}{2} \).
• The fourth term: Again, multiply the previous term by the common ratio, \( \frac{3}{2} \times -\frac{1}{4} = -\frac{3}{8} \).
• The fifth term: Follow the pattern for one more term, \( -\frac{3}{8} \times -\frac{1}{4} = \frac{3}{16} \).

By following this pattern, you can easily find the first five terms of a geometric sequence.

common ratio

The common ratio is the backbone of a geometric sequence. It's the constant factor that each term is multiplied by to produce the next term in the series. In our example, the common ratio is given as \(-\frac{1}{4}\).

Understanding the role of the common ratio is crucial:

• A positive common ratio means the sequence terms will always have the same sign if the first term is positive.
• A negative common ratio, like our example, causes the signs of the terms to alternate between positive and negative.
• If the common ratio is greater than 1, the sequence will grow larger with each term.
• If the common ratio is between -1 and 1, like \(-\frac{1}{4}\), the terms will decrease in absolute value.

With a firm grasp of the common ratio, you can predict how the sequence behaves and generates new terms.

sequence formula

A special formula helps in finding any term of a geometric sequence without listing all previous terms. This is particularly helpful for sequences with many terms.

For a geometric sequence, the n-th term \(a_n\) can be calculated using the formula:\[a_n = a_1 \, \cdot \, r^{n-1}\]Where:

• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the term number.

Using our original exercise, to find the second term, we set \(n = 2\):\[a_2 = 6 \, \cdot \, \left(-\frac{1}{4}\right)^{2-1} = -\frac{3}{2}\]This formula saves time and energy, allowing you to quickly calculate any term in the sequence directly. It's a powerful tool for mastering geometric sequences.