Problem 19 - Algebra and Trigonometry

Question

In Exercises \(17-24,\) 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. $$ 18,6,2, \frac{2}{3}, \dots $$

Step-by-Step Solution

Answer

The seventh term (\(a_7\)) of the given geometric sequence is \( \frac{2}{9} \).

Step 1: Identify the common ratio

The common ratio (r) can be found by dividing any term in the sequence by the preceding term. For this sequence, it can be found by dividing the second term (6) by the first term (18). Therefore \(r = \frac{6}{18} = \frac{1}{3}\).

Step 2: Write the formula for the nth term

The formula for the nth term of geometric sequence is used here with a as the first term (18) and r as the common ratio we obtained in step 1 which is \(\frac{1}{3}\). The formula is \(a_n = a \cdot r^{(n-1)}\). So, it becomes \(a_n = 18 \cdot (\frac{1}{3})^{(n-1)}\)

Step 3: Find the seventh term

Now, substituting n = 7 in the formula from step 2 to find the seventh term, \(a_7 = 18 \cdot (\frac{1}{3})^{(7-1)} = 18 \cdot (\frac{1}{3})^6 = \frac{2}{9}\).

Key Concepts

Common Ratio

A geometric sequence is defined by its ability to multiply each term by a constant value to get the next term. This constant is known as the common ratio. To find the common ratio in a sequence, you divide any term by the term preceding it. For instance, in the sequence 18, 6, 2, \( \frac{2}{3} \), the common ratio \( r \) can be calculated by dividing the second term (6) by the first term (18). This results in \( r = \frac{6}{18} = \frac{1}{3} \).
This means each term is \( \frac{1}{3} \) of the previous term, indicating a decrease. Identifying the common ratio is essential for understanding the pattern in a geometric sequence and for developing the nth term formula.

Nth Term Formula

To determine any term in a geometric sequence quickly, the nth term formula is utilized. The formula is given by: \( a_n = a \cdot r^{(n-1)} \). Here, \( a \) represents the first term in the sequence, and \( r \) is the common ratio. For our sequence, the first term \( a \) is 18, and the common ratio \( r \) is \( \frac{1}{3} \).

• The formula now becomes \( a_n = 18 \cdot \left(\frac{1}{3}\right)^{(n-1)} \).
• This formula helps to predict any term without needing all the preceding terms.

Simply substitute the term number into \( n \) and solve. Understanding how to apply this formula gives you flexibility in dealing with geometric sequences.

Seventh Term Calculation

Calculating the seventh term \( a_7 \) of any geometric sequence using the nth term formula provides a practical example of its application. First, you take the formula \( a_n = 18 \cdot \left(\frac{1}{3}\right)^{(n-1)} \). Then, substitute 7 for \( n \):
\[ a_7 = 18 \cdot \left(\frac{1}{3}\right)^{(7-1)} \]
This simplifies to:
\[ a_7 = 18 \cdot \left(\frac{1}{3}\right)^6 \]
After performing the arithmetic, it results in:
\[ a_7 = \frac{2}{9} \]
This means the seventh term is \( \frac{2}{9} \), verifying the general formula's accuracy. By understanding these calculations, you can grasp how each term is just a step further along in the sequence, shrinking consistently by the common ratio.