Problem 22 - 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. $$5,-1, \frac{1}{5},-\frac{1}{25}, \dots$$

Step-by-Step Solution

Answer

The formula for the nth term of the geometric sequence is \(a_n = 5 * (-0.2)^{n-1}\) and the seventh term of the series is 0.00032.

Step 1: Identification of parameters

Start by identifying the first term \(a_1\) and the common ratio \(r\). In this sequence, the first term \(a_1\) is 5 and the common ratio \(r\) can be found by dividing the second term by the first term, \(-1 / 5 = -0.2\).

Step 2: Formulation of the nth term formula

The formula for the nth term of a geometric sequence is \(a_n = a_1 * r^{(n-1)}\). Substituting for \(a_1 = 5\) and \(r = -0.2\), the formula becomes \(a_n = 5 * (-0.2)^{n-1}\).

Step 3: Calculation of the 7th term

To find the 7th term, plug in \(n = 7\) into the formula, \(a_7 = 5 * (-0.2)^{7 - 1} = 5 * (-0.2)^6 = 5 * (0.000064) = 0.00032\).

Key Concepts

nth term formula

The nth term formula is essential in identifying any particular term in a geometric sequence without listing all preceding terms. This formula is expressed as follows: \[ a_n = a_1 \times r^{(n-1)} \] Here, \(a_n\) represents the nth term you wish to find, \(a_1\) is the first term of the sequence, and \(r\) is the common ratio. Let's break it down:

• The expression \(a_1\) is just the first term of the sequence we started with.
• \(r^{(n-1)}\) indicates the number of times we multiply by the common ratio to reach the nth term from the first term.

In practice, if you have the sequence 5, -1, 1/5, -1/25, \(a_1\) would be 5, making your calculations much simpler. This formula saves time and effort by allowing a direct calculation of any term in the sequence.

common ratio

A critical component of any geometric sequence is the common ratio. It defines how the sequence progresses from one term to the next. To find the common ratio \(r\), you divide any term in the sequence by its preceding term. Taking the earlier sequence: 5, -1, \(\frac{1}{5}\), -\(\frac{1}{25}\):

• The common ratio \(r\) is calculated by taking \(-1 \div 5 = -0.2\).
• You can further verify it by checking the division of subsequent terms; \(\frac{1}{5} \div (-1) = -0.2\).

The consistency of this ratio is key to confirming that you indeed have a geometric sequence. Moreover, it dictates the formula you use to find other terms, as it captures the very relationship between terms in the sequence.

geometric sequence examples

Understanding geometric sequences becomes clearer through examples. Let's revisit the sequence given in the original problem: 5, -1, \(\frac{1}{5}\), -\(\frac{1}{25}\). This sequence has a clear pattern, which is the hallmark of a geometric sequence. Starting with 5 as the first term:

• The second term is obtained by multiplying the first term with the common ratio: \(5 \times -0.2 = -1\).
• Continuing this: \(-1 \times -0.2 = \frac{1}{5}\)
• And again: \(\frac{1}{5} \times -0.2 = -\frac{1}{25}\)

This step-by-step multiplication by the common ratio (-0.2) illustrates the geometric sequence in action. Such calculations verify that we're correctly applying the nth term formula, offering a reliable way to predict future terms. As you practice with various sequences, you'll enhance your intuitive understanding of geometric sequences and their properties.