Problem 28 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{4}=16, r=0.5, n=8 $$

Step-by-Step Solution

Answer

The 8th term is 1.

Step 1: Identify the Formula for the nth Term

In a geometric sequence, the nth term is given by the formula: \[ a_n = a_1 imes r^{(n-1)} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Calculate the First Term

We know the 4th term, \( a_4 = 16 \), and \( r = 0.5 \). Using the formula for the nth term, we can calculate the first term \( a_1 \):\[ a_4 = a_1 imes r^{(4-1)} \]\[ 16 = a_1 imes (0.5)^3 \]\[ 16 = a_1 imes 0.125 \]\[ a_1 = \frac{16}{0.125} \]\[ a_1 = 128 \]

Step 3: Use the Formula to Find the 8th Term

Now that we have \( a_1 = 128 \), we can find the 8th term (\( a_8 \)) using the formula:\[ a_8 = a_1 imes r^{(8-1)} \]\[ a_8 = 128 imes (0.5)^7 \]\[ a_8 = 128 imes 0.0078125 \]\[ a_8 = 1 \]

Key Concepts

Nth Term Formula

When dealing with geometric sequences, one important tool is the formula for the nth term. This formula helps us find any term within the sequence without needing to list every term. The formula is:\[ a_n = a_1 \times r^{(n-1)} \]Where:

• \( a_n \) is the nth term we're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio between consecutive terms.
• \( n \) is the position of the term in the sequence.

To use this formula, you simply substitute the known values and solve for the unknown. It’s particularly useful because it avoids the tedious task of computing all previous terms. Instead, you jump directly to the term you need!

Common Ratio

The common ratio is a fundamental feature of geometric sequences. It describes the factor by which we multiply one term to get to the next. In our example, the common ratio \( r \) is given as 0.5.Here's how the common ratio works:

• If the common ratio is greater than 1, the terms in the sequence will grow larger.
• If the common ratio is between 0 and 1, as in this case, the terms will shrink.
• If the common ratio is negative, the sequence will alternate in sign.

The common ratio is crucial because it tells us the nature of growth or decay in the sequence. It’s calculated by dividing any term by the previous one, which makes it simple to find once you know two consecutive terms.

First Term

Finding the first term \( a_1 \) is often necessary to solve for other terms in a geometric sequence. In our exercise, the first term was not directly given, so we had to calculate it using the 4th term and the common ratio.To solve for \( a_1 \), we rearrange the nth term formula:Given \( a_4 = 16 \) and \( r = 0.5 \), the formula was:\[ a_4 = a_1 \times r^{(4-1)} \]By isolating \( a_1 \), we find:\[ a_1 = \frac{16}{0.125} = 128 \]The first term acts like a starting point for any geometric sequence. Once it's known, and with knowledge of the common ratio, any subsequent term can be calculated using the nth term formula.