Problem 22 - 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. $$ 5,-1, \frac{1}{5},-\frac{1}{25}, \dots $$

Step-by-Step Solution

Answer

Using the formula, we calculate that the seventh term of the sequence \(a_7\) is \(-0.000128\).

Step 1: Identify the first term and the common ratio

The first term \(a_1\) of the sequence is 5. The common ratio (r) can be found by dividing the second term by the first term, and this should be the same for all consecutive terms in the sequence. Here, \(r = -1 / 5 = -0.2\)

Step 2: Write down the formula for the nth term

The formula for the nth term \(a_n\) of a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\). Here, our formula becomes \(a_n = 5 \cdot (-0.2)^{(n-1)}\)

Step 3: substitute n=7 into the formula

Substitute \(n = 7\) into the formula. So we have \(a_7\) = 5 * \((-0.2)^{7-1}\)

Key Concepts

nth term formula

When learning about geometric sequences, one of the first things you need to know is how to find any term quickly. This is where the nth term formula comes into play. For geometric sequences, the nth term formula is expressed as: \[ a_n = a_1 imes r^{(n-1)} \]where:

• \(a_n\) is the nth term you're solving for.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio or the factor by which each term is multiplied to get the next.
• \(n\) is the term number you want to find.

In simpler terms, the nth term formula involves taking the first term and multiplying it by the common ratio raised to the power of one less than the desired term's position. This formula lets you dive directly to any position in the sequence.

common ratio

In any geometric sequence, one of the most critical components is the common ratio. The common ratio is the factor that allows each term in the sequence to be derived from the preceding one. To determine the common ratio \(r\), you simply divide one term by the previous term. For example, in the given sequence:

• From 5 to -1: \( r = \frac{-1}{5} \)
• From -1 to \(\frac{1}{5}\): \( r = \frac{1/5}{-1} \)

Each time, you should find the same value, revealing the common ratio that dominates the sequence. This consistency is key because it confirms that the sequence is indeed geometric. Understanding the common ratio is foundational, as it directly influences the development of the nth term.

geometric progression

Now, let's talk about what makes a sequence geometric, often referred to as geometric progression. This type of sequence is fundamentally characterized by its constant ratio between consecutive terms. Geometric progressions can be either increasing or decreasing:

• Increasing: when \(r > 1\), each term gets larger.
• Decreasing: when \(0 < r < 1\), each term gets smaller.
• Alternating: when \(r < 0\), the terms alternate between positive and negative values, causing the sequence to oscillate.

Our original sequence (5, -1, \(\frac{1}{5}\), -\(\frac{1}{25}\), ...) shows an alternating progression with a decreasing magnitude. The common ratio \(-0.2\) makes terms switch signs and decrease, highlighting the diversity within geometric progression types.

sequence analysis

Sequence analysis involves breaking down the terms and behaviors of a sequence to better understand its structure and predict future values. For a geometric sequence, you'll often focus on components like the initial term, the common ratio, and how each term changes compared to others. To analyze the sequence efficiently, follow these steps:

• Identify the first term and common ratio.
• Apply the nth term formula to determine subsequent terms.
• Analyze how the sequence develops: is it shrinking, growing, or alternating?
• Create a possible graph of the sequence to visualize behaviors.

These steps provide a framework for dissecting the nuances of geometric sequences, allowing you to address exercises more strategically. Understanding these elements lays the groundwork for deeper mathematical exploration and better equips you to solve complex problems.