Problem 20 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$12,6,3, \frac{3}{2}, \dots$$

Step-by-Step Solution

Answer

The formula for the nth term of the sequence is \(a_{n} = 12 * 0.5^{(n-1)}\), and the seventh term of the sequence is \(a_{7} = 0.1875\)

Step 1: Find the Common Ratio

To find the common ratio, divide the second term by the first term. So \(r = \frac{6}{12} = 0.5\)

Step 2: Write the formula for the nth term

Now that we know the first term \(a_{1} = 12\) and the common ratio \(r = 0.5\), we can use the general formula for the nth term of a geometric sequence \(a_{n} = a_{1} * r^{(n-1)}\). This gives us \(a_{n} = 12 * 0.5^{(n-1)}\)

Step 3: Find the seventh term

To find the seventh term, substitute \(n = 7\) into the formula: \(a_{7} = 12 * 0.5^{(7-1)} = 12 * 0.5^{6} = 12 * 0.015625 = 0.1875\)