Problem 58 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the geometric sequence. $$a_{1}=10, a_{2}=2$$

Step-by-Step Solution

Answer

The general term is \(a_n = 10 \cdot (0.2)^{n-1}\).

Step 1: Identify the First Term

The problem states that the first term of the geometric sequence is given as \(a_1 = 10\). This is our starting point for the sequence.

Step 2: Calculate the Common Ratio

To find the common ratio \(r\) of a geometric sequence, divide the second term \(a_2 = 2\) by the first term \(a_1 = 10\). Thus, \(r = \frac{a_2}{a_1} = \frac{2}{10} = 0.2\).

Step 3: Write the General Term Formula

The general term of a geometric sequence is given by the formula \(a_n = a_1 \cdot r^{n-1}\). We already know \(a_1 = 10\) and \(r = 0.2\). Substitute these values into the formula: \(a_n = 10 \cdot (0.2)^{n-1}\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the **common ratio**. This ratio is constant throughout the sequence. It helps in identifying how the sequence progresses. If the ratio is greater than one, the terms **grow exponentially**. Conversely, if it's between zero and one, the terms **shrink**. To find the common ratio, you divide the second term of the sequence by the first term. Formally, for a sequence with the first term as \(a_1\) and second term as \(a_2\), the common ratio \(r\) is given by:

• \(r = \frac{a_2}{a_1}\)
• For the given sequence, \(r = \frac{2}{10} = 0.2\).

This calculation shows how each term is derived from its predecessor. Understanding the common ratio is vital as it feeds directly into other calculations involving the sequence, such as finding unknown terms or determining the behavior of the sequence over time.

The General Term Formula Explained

The **general term formula** of a geometric sequence provides a way to compute any term in the sequence from its position alone. This formula is extremely useful as it reduces the need to manually multiply the terms by the common ratio repeatedly to reach a specific position.In mathematical terms, if you know the first term \(a_1\) and the common ratio \(r\), the formula for the \(n\)-th term \(a_n\) is:

• \(a_n = a_1 \cdot r^{n-1}\)

This formula hinges on two parts:

• \(a_1\), the initial value or starting point of the sequence.
• \(r^{n-1}\), which represents the compounding effect of the ratio applied over \(n-1\) steps.

For the given sequence, with \(a_1 = 10\) and \(r = 0.2\), substituting these values into the formula gives:

• \(a_n = 10 \cdot (0.2)^{n-1}\)

This equation allows the calculation of any term simply by inputting the desired term position \(n\). It encapsulates the essence of geometric growth or decay within the sequence.

Sequence Calculation Made Simple

Calculating terms in a geometric sequence involves using the general term formula. Armed with knowledge of the common ratio and the first term, you can find any term in the sequence quickly.Here's the simple process for using the sequence calculation to find any term. Imagine you want to find the term at position \(n = 4\):

• Start with the known general term formula: \(a_n = a_1 \cdot r^{n-1}\)
• Substitute known values: For the fourth term, \(n=4\), \(a_1 = 10\), and \(r = 0.2\).
• Calculate: \(a_4 = 10 \cdot (0.2)^{4-1}\)
• Evaluate the exponent: \((0.2)^3 = 0.008\)
• Final computation: \(a_4 = 10 \cdot 0.008 = 0.08\)

By following these steps, you can calculate any term within the sequence effortlessly. It saves time and avoids repetitive multiplication, showcasing the power and efficiency of using algebraic formulas in sequence calculations. This method also helps detect patterns or significant changes as you proceed through the sequence, providing deeper insight into its structure.