Problem 20 - Algebra and Trigonometry

Question

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 12,6,3, \frac{3}{2}, \dots $$

Step-by-Step Solution

Answer

The formula for the nth term of the given geometric sequence is \(a_{n} = 12 * (1/2)^{n - 1}\). The 7th term of this sequence, \(a_{7}\), is 0.1875.

Step 1: Find the common ratio

Observe the given sequence, it can be noticed that each term is half of the previous term, which gives us the common ratio(r) = 1/2.

Step 2: Write down the general formula for the nth term

The formula for the nth term of a geometric sequence is given by \(a_{n} = a_{1} * r^{(n - 1)}\). Substituting the known values, we get \(a_{n} = 12 * (1/2)^{n - 1}\). This is the formula for any term in the sequence.

Step 3: Find the 7th term of the sequence

Use the formula from step 2 to find the 7th term of the sequence by substituting n = 7. So, \(a_{7} = 12 * (1/2)^{7 - 1} = 12 * (1/2)^6 = 12 * (1/64) = 0.1875\).

Key Concepts

nth term formula

In a geometric sequence, the "nth term formula" is a powerful tool that allows you to determine the value of any term within the sequence. This formula is expressed as \(a_{n} = a_{1} \times r^{(n-1)}\). It effectively uses the initial term, represented by \(a_{1}\), and the common ratio \(r\), along with the position \(n\) of the term you are looking for. The formula helps in effortlessly pinpointing the specific term within the sequence, ensuring you don't have to redundantly list out every prior term.
For example, if you need to find the 10th term in a sequence starting with 3, with a common ratio of 2, you simply plug into the formula: \(a_{n} = 3 \times 2^{(10-1)}\), resulting in the precise 10th term.
Understanding the nth term formula allows you to dissect any sequence with ease, saving time and eliminating unnecessary calculations.

common ratio

The **common ratio** is a fundamental aspect of a geometric sequence. It is the constant factor by which each term in the sequence is multiplied to arrive at the next term. Understanding and identifying the common ratio is essential for unraveling the pattern of a geometric sequence.
Let's break it down with an example: consider the sequence 12, 6, 3, \(\frac{3}{2}\), etc. To find the common ratio, divide the second term by the first term: \(r = \frac{6}{12} = \frac{1}{2}\). In this case, the common ratio \(r\) is \(\frac{1}{2}\), meaning each term is half of the preceding term.
In any sequence, once you identify the common ratio, it becomes the key to constructing the nth term formula and finding various terms within the sequence. The common ratio helps to maintain the consistent structure of the sequence, providing a clear, predictable pattern to follow.

general term

The **general term** in a geometric sequence is essentially another way of referring to the nth term. It represents a formula that can be used to find any specific term in the sequence. This term comprises the initial term \(a_{1}\), the common ratio \(r\), and the term number \(n\). Understanding the general term formula \(a_{n} = a_{1} \times r^{(n-1)}\) enables you to predict and calculate any term in the sequence efficiently.
Consider a scenario where a sequence begins with 5, and the common ratio is 3. To find the general term \(a_{n}\), you would set up the formula as \(a_{n} = 5 \times 3^{n-1}\).
This formula is powerful, as it allows you to insert various values for \(n\) to determine any term within the sequence, offering a comprehensive understanding of how the sequence grows or decreases over time.

sequence formula

The **sequence formula** for a geometric sequence elegantly encapsulates the entire sequence into one streamlined equation, \(a_{n} = a_{1} \times r^{(n-1)}\). This compact expression is important for anyone needing to generate terms in a sequence or pinpoint any term without enumerating them all.
When dealing with a sequence like 4, 8, 16, ..., it follows the formula with an initial term \(a_{1} = 4\) and a common ratio \(r = 2\). The sequence formula consequently becomes \(a_{n} = 4 \times 2^{(n-1)}\).
By leveraging this formula, you can swiftly compute any term in the sequence. It's an invaluable method for predicting the sequence's pattern and growth progression, allowing for the exploration of complex sequence properties efficiently and accurately.