Problem 33 - Beginning and Intermediate Algebra

Question

Find the indicated term of each geometric sequence. $$27,-9,3,-1, \dots ; a_{8}$$

Step-by-Step Solution

Answer

The eighth term in the given geometric sequence is \(-\frac{1}{81}\).

Step 1: Identify the first term and common ratio

Denote the first term as \(a_1\) and the common ratio as \(r\). By observing the sequence, we notice the first term is 27 and the common ratio can be found by dividing the second term by the first term \((-9 / 27 = -1 / 3)\). Therefore, we have \(a_1 = 27\) and \(r = -\frac{1}{3}\).

Step 2: Use the formula for the nth term

The formula for the nth term of a geometric sequence is given by: \[a_n = a_1 \times r^{n-1}\] Now, we want to find the 8th term, so we substitute \(n = 8\), \(a_1 = 27\), and \(r = -\frac{1}{3}\) into the formula.

Step 3: Calculate the 8th term

Using the formula: \[a_8 = 27 \times \left(-\frac{1}{3}\right)^{8-1}\] Evaluate the exponent: \[a_8 = 27 \times \left(-\frac{1}{3}\right)^7\] Now, multiply 27 by the result: \[a_8 = 27 \times \frac{-1}{2187}\] After simplifying the expression: \[a_8 = -\frac{27}{2187}\]

Step 4: Simplify the result

Simplify the fraction: \[a_8 = -\frac{1}{81}\]

Step 5: Conclusion

The eighth term in the given geometric sequence is \(-\frac{1}{81}\).

Key Concepts

The nth Term Formula

The concept of the nth term formula is crucial in understanding how geometric sequences work. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, known as the "common ratio." This formula generally helps identify any term in a sequence, without listing all the preceding terms.
To find the nth term in a geometric sequence, we use the formula:

• \( a_n = a_1 \times r^{n-1} \)

Where:

• \( a_n \) is the nth term we're looking for.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number we want to find.

By substituting the corresponding values into this formula, we can calculate the desired term directly. This method is effective and saves time, especially for sequences with many terms.

Understanding the Common Ratio

In any geometric sequence, the common ratio \( r \) is the factor by which we multiply each term to obtain the next term. It plays a key role in forming the sequence and is calculated by dividing any term by its preceding term.
For example, if the sequence is 27, -9, 3, -1, ..., you find \( r \) by dividing the second term, -9, by the first term, 27:

• \( r = \frac{-9}{27} = -\frac{1}{3} \)

This ratio remains constant throughout the sequence and helps determine how the sequence progresses.
Recognizing the common ratio in a sequence makes it easier to understand how each term relates to the others and is fundamental for using the nth term formula effectively. If \( r \) is a fraction, the sequence will decrease in magnitude, whereas if it exceeds 1, the sequence will grow without bound. Hence, mastering the calculation of the common ratio equips you to handle any geometric sequence with confidence.

Sequence Simplification

Simplifying terms in a sequence is often necessary to obtain more manageable numbers. In particular, geometric sequences can produce very large or very small numbers whose terms can be simplified for better comprehension.
In our example, after calculating the 8th term of the sequence, we obtained \(-\frac{27}{2187}\). This fraction might initially seem complex, but it can be simplified by determining the greatest common divisor of the numerator and the denominator.
Here, both 27 and 2187 share a common factor of 27:

• \( \frac{-27}{2187} = -\frac{1}{81} \)

Simplifying the result ensures clarity and aids in understanding the sequence's progression. It also allows for easier communication and use of the answer in further calculations or reasoning. Keeping terms simplified makes working with sequences considerably more efficient and straightforward, particularly when dealing with intricate mathematical operations.