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 general term of the sequence is \(a_{n} = 5 * (-1/5)^{n-1}\), and the seventh term of the sequence is \(a_{7} = -1/15625\).

Step 1: Identify the common ratio

Observe the sequence to determine the common ratio \(r\). This can be found by dividing any term by the preceding term. For instance, dividing the second term (-1) by the first term (5) gives \(r = -1/5\). Similarly, dividing the third term (\(1/5\)) by the second term (-1) also results in \(r = -1/5\). We verify that this is the same for all terms, so we can confirm that -1/5 is the common ratio of this geometric sequence.

Step 2: Write the formula for the general term

The general formula for the \(n^{th}\) term of a geometric sequence is \(a_{n} = a_{1} * r^{n-1}\), where \(a_{n}\) stands for the \(n^{th}\) term, \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. For this geometric sequence, the first term \(a_{1} = 5\), and the common ratio \(r = -1/5\). Therefore, the general term of this sequence is \(a_{n} = 5 * (-1/5)^{n-1}\).

Step 3: Find the seven term of the sequence

With the general formula \(a_{n} = 5 * (-1/5)^{n-1}\), substitute \(n = 7\) to find the seventh term of the sequence: \(a_{7} = 5 * (-1/5)^{7-1} = 5 * (-1/5)^{6}\). Therefore, \(a_{7} = -1/15625\).