Problem 34 - Beginning and Intermediate Algebra

Question

Find the indicated term of each geometric sequence. $$-\frac{1}{125},-\frac{1}{25},-\frac{1}{5},-1, \ldots ; a_{7}$$

Step-by-Step Solution

Answer

The 7th term of the geometric sequence \(a_n = -\frac{1}{125} \times 5^{n-1}\) is \(a_7 = -125\).

Step 1: Identify the common ratio

To determine the common ratio, divide any term of the sequence by its previous term: $$r = \frac{-\frac{1}{25}}{-\frac{1}{125}} = \frac{1}{25} \times \frac{125}{1} = 5$$ So the common ratio is \(5\).

Step 2: Determine the general formula of the geometric sequence

Now, let's find the general formula of the geometric sequence. The formula for any term in a geometric sequence is: $$a_n = a_1 \times r^{n-1}$$ where \(n\) is the term number, \(a_1\) is the first term, and \(r\) is the common ratio. In our case, the first term \(a_1 = -\frac{1}{125}\) and the common ratio \(r = 5\).

Step 3: Calculate the 7th term using the general formula

Next, let's find the 7th term by plugging the values of \(a_1\), \(r\), and \(n\) into the general formula: $$a_7 = -\frac{1}{125} \times 5^{7-1}$$

Step 4: Simplify the expression

Finally, we simplify the expression to determine the value of the 7th term: $$a_7 = -\frac{1}{125} \times 5^6 = -\frac{1}{125} \times 15625 = -125$$ Thus, the 7th term of the given geometric sequence is \(-125\).

Key Concepts

Common Ratio

In a geometric sequence, understanding the common ratio is key to unlocking the sequence's pattern. The common ratio is a consistent factor that you multiply by to move from one term to the next. To find the common ratio, you select any term in the sequence and divide it by the term that comes just before it. For example, in the sequence given:

• Start with the second term \( -\frac{1}{25} \) and divide it by the first term \( -\frac{1}{125} \).
• Mathematically, we can express the common ratio as \left( r = \frac{-\frac{1}{25}}{-\frac{1}{125}} = 5 \right)\.

This consistent multiplication of five helps maintain the pattern in our sequence.

General Formula

The general formula is your magic tool that gives you the key to understanding any term in a geometric sequence. Once you have the first term and the common ratio, you can predictably calculate the value of any term. The general formula for a geometric sequence is:

• \(a_n = a_1 \times r^{n-1}\)

Here:

• \(a_1\) is the initial or first term of the sequence.
• \(r\) is your common ratio.
• \(n\) represents the position of the term you’re interested in.

For our sequence, \(a_1 = -\frac{1}{125}\) and \(r = 5\). Thus, you have a formula that allows you to calculate any term if you know \(n\), making problem-solving simpler and the sequence more manageable.

Sequence Term Calculation

Now that we understand the common ratio and have the general formula, it's easy to calculate any term in the sequence. To find the 7th term, you simply plug your values into the general formula:

• First, note that \(n = 7\), \(a_1 = -\frac{1}{125}\), and \(r = 5\).

Using the formula:

• \(a_7 = -\frac{1}{125} \times 5^{7-1}\)

That powers the common ratio to 6, which is one less than the term number:

• \(a_7 = -\frac{1}{125} \times 15625\).

Finally, simplify to find the 7th term:

• \(a_7 = -125\)

In this manner, the 7th term of the sequence is calculated, ensuring clarity and accuracy in the result.