Problem 49 - Algebra 2

Question

Find the 10 th term of each geometric sequence. $$ a_{11}=8, r=\frac{1}{2} $$

Step-by-Step Solution

Answer

The 10th term of the geometric sequence is 4.

Step 1: Figure out the first term

Using the formula \( a_{n} = a_{1} * r^{(n-1)} \), we can substitute the given 11th term \( a_{11} = 8 \) and the common ratio \( r = \frac{1}{2} \) to find out the first term of the sequence. This means we can set up the equation as follows: \( 8 = a_{1} * (\frac{1}{2})^{(11-1)} \). Solving for \( a_{1} \) gives \( a_{1} = 8 / (\frac{1}{2})^{10} \), which simplifies to \( a_{1} = 2048 \).

Step 2: Find the 10th term

With the first term \( a_{1} = 2048 \) and the common ratio \( r = \frac{1}{2} \), we can now find the 10th term using the same formula: \( a_{10} = a_{1} * r^{(10-1)} \). This will give \( a_{10} = 2048 * (\frac{1}{2})^{9} \). Solving this will give us the 10th term.

Step 3: Calculate the 10th term

Solving the equation from Step 2 will give us \( a_{10} = 2048 * (\frac{1}{2})^{9} = 4 \). Therefore, the 10th term of the sequence is 4.

Key Concepts

Term Formulas

In any geometric sequence, each term is found by using a specific type of formula called a **term formula**. This formula allows us to determine any term in the sequence, based on the first term and the common ratio. The general term formula for a geometric sequence is given by:\[a_{n} = a_{1} \times r^{(n-1)}\]Where:

• \(a_{n}\) is the term we want to find.
• \(a_{1}\) is the first term of the sequence.
• \(r\) is the common ratio, a key characteristic of geometric sequences.
• \(n\) is the term number we are interested in.

Using this formula, students can solve for any missing term in a geometric sequence. It's particularly useful when we already know the value of a particular term and need to reverse-calculate or find other terms.

Common Ratio

The **common ratio** is a pivotal component of any geometric sequence. It is the factor between each term and the next in the sequence. When multiplying the current term by this ratio, you can find the next term. In our specific case, the common ratio \(r\) is \(\frac{1}{2}\).To identify or verify the common ratio in a sequence:

• Select two consecutive terms from the sequence, such as \(a_{n}\) and \(a_{n+1}\).
• Divide the second term by the first: \(r = \frac{a_{n+1}}{a_{n}}\).

This ensures uniformity across the sequence, meaning each term grows or decreases at a constant rate. Recognizing the common ratio simplifies understanding how geometric sequences behave, especially when solving for unspecified terms.

Sequence Calculation

Calculating particular terms within a geometric sequence can seem daunting at first, yet it's straightforward with a solid grasp of the basic formula and calculations involved.For instance, if you're asked to find the 10th term and you already know:

• The first term, \(a_1\), which we've calculated to be 2048.
• The common ratio, \(r\), which is \(\frac{1}{2}\).

The calculation involves plugging these values into the term formula:\[a_{10} = a_{1} \times r^{(n-1)} = 2048 \times \left(\frac{1}{2}\right)^{9}\]From here, it's a matter of performing the arithmetic. Computing \((\frac{1}{2})^{9}\) and then multiplying by 2048 will yield the 10th term of 4.Understanding the calculation steps not only helps in solving problems but also deepens comprehension of how geometric progressions operate systematically.