Problem 14 - Algebra and Trigonometry

Question

In Exercises \(9-16,\) use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) Find \(a_{30}\) when \(a_{1}=8000, r=-\frac{1}{2}\)

Step-by-Step Solution

Answer

After step 3, the simplified value will be your answer to the exercise. Always remember to adhere to arithmetic operations rules.

Step 1: Identify the parameters

Here, the first term (\(a_{1}\)) is 8000, the common ratio (r) is -1/2 and the term to find (\(n\)) is 30.

Step 2: Apply the formula for the nth term of a geometric sequence

Substitute the given parameters into the formula \(a_{n} = a_{1} * r^{(n-1)}\). So, \(a_{30} = 8000 * (-\frac{1}{2})^{(30-1)}\)

Step 3: Simplify

Simplify the calculation to get the result. Ensure proper use of arithmetic rules involving operations of negative numbers and powers.

Key Concepts

nth term formula

The nth term formula in a geometric sequence serves as a tool to find any specific term without listing the entire sequence. A geometric sequence multiplies each term by a consistent number, known as the common ratio, to obtain the next term. The general expression to find the nth term \(a_n\) in a geometric sequence is given by: \[a_n = a_1 \times r^{(n-1)}\] Where:

• \(a_1\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the term number you are trying to find.

To solve problems like finding the 30th term of a sequence, this formula is your go-to method. You substitute the known values into the formula and proceed with calculations. The use of this formula saves time and simplifies problems by allowing us to find the value of a term directly rather than computing each preceding term. To deepen your understanding, always ensure you understand how the formula is applied through examples.

common ratio

The common ratio is a key characteristic of a geometric sequence and denotes the factor by which you multiply each term to get the next one. In the context of this sequence, where the first term is 8000 and the common ratio is \(-\frac{1}{2}\), it means each term is half the previous term and negative. The presence of a negative common ratio introduces a change in the sign of the sequence’s terms, causing them to oscillate between positive and negative.

• If \(|r| < 1\), the terms decrease in absolute value over time.
• If \(r > 0\), the sequence remains positive or negative throughout.
• If \(r < 0\), the sequence alternates in sign.

Understanding the common ratio is fundamental, as it dictates the pattern and direction of the sequence's growth or decay. Practicing with different common ratios will solidify your grasp of how they influence sequences.

first term in sequences

The first term of a sequence, represented as \(a_1\), is where the sequence begins, setting the stage for all subsequent values. In this example, the first term is given as 8000. Setting an initial term is crucial as it serves as the foundation for calculating all other terms in the sequence using the nth term formula. The first term:

• Identifies the starting point of the sequence.
• Helps in directly plugging into the nth term formula.
• Affects the values of all terms that follow it.

Moreover, understanding how the first term contributes to the sequence’s development can help identify distinct features and boundaries within the sequence. Each shift in the first term transforms the whole sequence, demonstrating the dynamic nature of geometric sequences.