Problem 6 - Precalculus: Mathematics for Calculus

Question

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$a_{n}=6(-0.5)^{n-1}$$

Step-by-Step Solution

Answer

(a) First five terms: 6, -3, 1.5, -0.75, 0.375. (b) Common ratio: -0.5. (c) Graph these values.

Step 1: Calculate the First Term

To find the first term, substitute \(n = 1\) into the given formula: \(a_1 = 6(-0.5)^{1-1}\). Simplify the expression to find \(a_1 = 6(-0.5)^0 = 6 \times 1 = 6\).

Step 2: Calculate the Second Term

Substitute \(n = 2\) into the formula: \(a_2 = 6(-0.5)^{2-1}\). Simplify the expression to get \(a_2 = 6(-0.5)^1 = 6 \times (-0.5) = -3\).

Step 3: Calculate the Third Term

Substitute \(n = 3\) into the formula: \(a_3 = 6(-0.5)^{3-1}\). Simplify to find \(a_3 = 6(-0.5)^2 = 6 \times 0.25 = 1.5\).

Step 4: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \(a_4 = 6(-0.5)^{4-1}\). Simplify to get \(a_4 = 6(-0.5)^3 = 6 \times (-0.125) = -0.75\).

Step 5: Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \(a_5 = 6(-0.5)^{5-1}\). Simplify to find \(a_5 = 6(-0.5)^4 = 6 \times 0.0625 = 0.375\).

Step 6: Identify the Common Ratio

The common ratio \(r\) in the sequence is the factor by which we multiply one term to get the next. From the formula \(a_n = 6(-0.5)^{n-1}\), we can see that the common ratio \(r\) is \(-0.5\).

Step 7: Graph the Terms

Plot the calculated terms on a graph with the horizontal axis representing the term index (1 to 5) and the vertical axis representing the term value (6, -3, 1.5, -0.75, 0.375). The sequence forms points that follow a geometric progression.

Key Concepts

Common Ratio

In the context of a geometric sequence, the common ratio is the constant factor that each term is multiplied by to get the next term in the sequence. It is denoted by the symbol \( r \). In the given formula \( a_n = 6(-0.5)^{n-1} \), the common ratio is identified directly within the formula as \(-0.5\).

Here's how it works:

• To move from the first term \(6\) to the second term \(-3\), multiply by \(-0.5\).
• To get from the second term \(-3\) to the third term \(1.5\), again multiply by \(-0.5\).
• This pattern continues for subsequent terms: third to fourth (\(1.5\) to \(-0.75\)), fourth to fifth (\(-0.75\) to \(0.375\)), by multiplying by \(-0.5\) each time.

The common ratio plays a crucial role in defining the geometric nature of the sequence, causing each term to shrink or expand based on whether \( |r| < 1 \), \( |r| = 1 \), or \( |r| > 1 \). In this case, \(|-0.5| < 1\), indicating that the terms decrease in absolute value.

Sequence Terms

The sequence terms are individual elements that make up the sequence, derived from the given sequence formula. Exploring the first five terms allows us to get a sense of how the sequence behaves. Let's break down these steps in detail:

- **First Term**: Substitute \(n = 1\) into the formula: \( a_1 = 6(-0.5)^{1-1} = 6 \times 1 = 6 \).- **Second Term**: Substitute \(n = 2\): \( a_2 = 6(-0.5)^{2-1} = 6 \times (-0.5) = -3 \).- **Third Term**: Substitute \(n = 3\): \( a_3 = 6(-0.5)^{3-1} = 6 \times 0.25 = 1.5 \).- **Fourth Term**: Substitute \(n = 4\): \( a_4 = 6(-0.5)^{4-1} = 6 \times (-0.125) = -0.75 \).- **Fifth Term**: Substitute \(n = 5\): \( a_5 = 6(-0.5)^{5-1} = 6 \times 0.0625 = 0.375 \).

These terms show the geometric nature as they alternate in sign and reduce in magnitude.

Graphing Sequences

Graphing a geometric sequence is an effective way to visualize how the terms change and interact over time. When plotting these terms derived from \( a_n = 6(-0.5)^{n-1} \), you will see a clear pattern emerge:

To create a graph:

• Begin by establishing your axes. The horizontal axis should represent the term number (from 1 to 5 in this case).
• The vertical axis will indicate the value of each term (such as 6, -3, 1.5, -0.75, 0.375).
• Mark each point on the graph by using the term number as the x-coordinate and the term value as the y-coordinate.

When plotted, these points will not form a smooth curve but rather a zig-zag pattern due to the changing sign and decreasing absolute values. This graphical representation helps to intuitively understand the alternating sign and diminishing trend which is characteristic of a sequence with a common ratio whose absolute value is less than 1.