Problem 11 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{3}=32, r=-0.5, n=6 $$

Step-by-Step Solution

Answer

The sixth term of the geometric sequence is -4.

Step 1: Understand the Geometric Sequence Formula

A geometric sequence follows the pattern \( a_n = a_1 \, r^{n-1} \), where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. We need to find \( a_6 \) given \( a_3 = 32 \), \( r = -0.5 \).

Step 2: Find the First Term Using the Third Term

We are given \( a_3 = 32 \) and \( r = -0.5 \). Use the formula for the \( n \)-th term to find the first term: \( 32 = a_1 (-0.5)^{2} \). Simplify to \( 32 = a_1 \times 0.25 \).

Step 3: Solve for the First Term

From \( 32 = a_1 \times 0.25 \), solve for \( a_1 \) by dividing both sides by \( 0.25 \): \( a_1 = \frac{32}{0.25} = 128 \).

Step 4: Calculate the Sixth Term Using the Formula

Now that we have \( a_1 = 128 \), use the sequence formula to find \( a_6 \): \( a_6 = 128 \, (-0.5)^{5} \).

Step 5: Simplify Exponent and Multiply

Calculate \((-0.5)^{5} = -0.03125 \). Then, multiply by \( 128 \): \( a_6 = 128 \, \times \, -0.03125 = -4 \).

Step 6: Verify the Calculation

Double check each calculation step. Recheck \( (-0.5)^{5} \) and verify multiplication to ensure \( a_6 = -4 \).

Key Concepts

Understanding the Geometric Sequence Formula

Geometric sequences are sequences where each term after the first is the product of the previous term and a constant, known as the common ratio. This relationship is captured in the geometric sequence formula: \[ a_n = a_1 imes r^{n-1} \] where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of the term in the sequence. This formula allows you to calculate any term in the sequence if you know the first term and the common ratio. It’s particularly useful because you don't need to know all the individual terms before it to find any specific term. To use this formula effectively:

• Identify \( a_1 \) and \( r \).
• Plug these values along with \( n-1 \) into the formula.
• Compute to find the \( n \)-th term.

Decoding the Common Ratio

The common ratio \( r \) is a crucial element in a geometric sequence. It determines how each term in the sequence multiplies to become the next term. Unlike in arithmetic sequences where you add a constant difference, here you multiply by this constant factor. The common ratio can be found if you are given two consecutive terms of the sequence. Simply divide the second term by the first. For example: - If you know that the first term \( a_1 \) is 128 and the second term is \( r \times a_1 \), divide the second term by \( a_1 \) to find \( r \). Remember:

• A positive \( r \) means all terms have the same sign as \( a_1 \).
• A negative \( r \) results in alternating positive and negative terms.

Understanding the common ratio can greatly simplify identifying the behavior and pattern of a sequence.

Finding the First Term

In many geometric sequence problems, the first term \( a_1 \) may not be given directly. Often, you will need to calculate it using other information, like a specific term and the common ratio. For example, if you know \( a_3 = 32 \) and \( r = -0.5 \), you can find the first term using the formula for the third term: \[ a_3 = a_1 imes (-0.5)^2 \] Solve this by substituting \( a_3 \) and simplifying: - \( 32 = a_1 imes 0.25 \) Then, to find \( a_1 \), divide each side by 0.25, resulting in: - \( a_1 = \frac{32}{0.25} = 128 \). This process highlights how versatile the geometric sequence formula can be in adjusting to different unknowns.

Calculation of the n-th Term

Finding the \( n \)-th term in a geometric sequence is straightforward using the geometric sequence formula once the first term \( a_1 \) and the common ratio \( r \) are known. Suppose you've determined:

• \( a_1 = 128 \)
• \( r = -0.5 \)

To find the sixth term \( a_6 \), use the formula: \[ a_6 = 128 imes (-0.5)^{5} \] Calculate \((-0.5)^{5}\) which equals \(-0.03125\), then multiply by 128: - \( a_6 = 128 imes -0.03125 = -4 \) Thus, the sixth term in this sequence is \(-4\). Practice finding the \( n \)-th term helps improve understanding and capability in recognizing sequence patterns quickly.