Problem 35 - Beginning and Intermediate Algebra

Question

Find the indicated term of each geometric sequence. $$-\frac{1}{64},-\frac{1}{32},-\frac{1}{16},-\frac{1}{8}, \dots, a_{12}$$

Step-by-Step Solution

Answer

The 12th term (\(a_{12}\)) in the geometric sequence is \(-32\).

Step 1: Find the common ratio

To find the common ratio between each term in this sequence, simply divide the second term by the first term or the third term by the second term to find the common ratio: \[ r=\frac{a_2}{a_1}=\frac{-\frac{1}{32}}{-\frac{1}{64}}=2 \]

Step 2: Use the geometric sequence formula to find the 12th term

The general formula for the nth term in a geometric series is: \[a_n = a_1 r^{n-1}\] In this case, we are looking for the 12th term (n=12), and we know the first term is \(-\frac{1}{64}\) (a_1 =\(-\frac{1}{64}\)) and the common ratio is 2 (r = 2). Replacing these values in the formula above, we have: \[ a_{12} = \left(-\frac{1}{64}\right) (2)^{12-1} \]

Step 3: Simplify the expression to find the 12th term

Now, we simplify the expression: \begin{align*} a_{12} = \left(-\frac{1}{64}\right) (2)^{11} &= \left(-\frac{1}{64}\right) (2048)\\ &= -\frac{2048}{64}\\ &= -32 \end{align*} So the 12th term in the geometric sequence is \(-32\).

Key Concepts

Common Ratio

In a geometric sequence, understanding the concept of the *common ratio* is key. It's the factor by which we multiply each term in the sequence to get the next one. To find this ratio, you can divide any term by the one preceding it. This helps maintain the consistency of the pattern across the sequence.
In the example given in the exercise, the sequence begins with the fractions:

• \(-\frac{1}{64} \)
• \(-\frac{1}{32} \)
• \(-\frac{1}{16} \)

To find the common ratio \( r \), you divide the second term by the first:\[r = \frac{-\frac{1}{32}}{-\frac{1}{64}} = 2\] This means each term is doubled to reach the subsequent term. Identifying the common ratio is crucial because it sets the pace and direction of the entire sequence.

Geometric Progression

A *geometric progression* is simply another way to say a geometric sequence. This is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Unlike arithmetic progression, which adds or subtracts a constant, geometric progressions multiply.

• The first term in our example is \(-\frac{1}{64}\), followed by repeating multiplication by the common ratio of 2.
• Given this consistency, every term can be calculated using this multiplication process.

Geometric progressions are not limited to positive terms or integer values, as demonstrated by our sequence of fractions and negative numbers. Recognizing and using this pattern is fundamental for efficiently solving problems involving geometric sequences.

Nth Term Formula

The *nth term formula* in a geometric sequence allows you to find any term in the sequence without writing all of the intermediate terms. The formula is expressed as:\[a_n = a_1 \cdot r^{n-1}\]Here's what each part means:

• \(a_n\) is the term you're looking for.
• \(a_1\) is the first term in the sequence.
• \(r\) is the common ratio.
• \(n\) is the position of the term you're finding, starting with the first term as position one.

Using the exercise example, to find the 12th term with \(a_1 = -\frac{1}{64}\), \(r = 2\), and \(n = 12\), it becomes:\[a_{12} = \left(-\frac{1}{64}\right) \cdot 2^{11}\]Simplifying this gives you the final value of \(-32\). Understanding and using this formula allows you to jump directly to any term in the sequence, saving time and ensuring accuracy.