Problem 30 - Beginning and Intermediate Algebra

Question

Find the general term of each geometric sequence. $$-\frac{1}{5},-\frac{3}{10},-\frac{9}{20},-\frac{27}{40}, \dots$$

Step-by-Step Solution

Answer

The general term of the geometric sequence is given by the formula: \(a_n = -\frac{1}{5} \cdot 3^{(n-1)}\).

Step 1: Identify the Common Ratio of the Sequence

To find the common ratio, divide the second term by the first term, the third term by the second term, and so on. \(\text{Common Ratio}(r) = \frac{-\frac{3}{10}}{-\frac{1}{5}} = \frac{3}{10} \cdot \frac{5}{1} = 3\) The common ratio for this geometric sequence is 3.

Step 2: Determine the General Term Formula

Now that we have identified the common ratio, we can use the formula for the nth term of a geometric sequence: $$a_n = a_1 \cdot r^{(n-1)}$$ where \(a_n\) is the nth term of the sequence, \(a_1\) is the first term, and \(r\) is the common ratio. In this case, the first term, \(a_1\) = -1/5 and the common ratio, \(r\) = 3. Now, plug in the values we have found: $$a_n = -\frac{1}{5} \cdot 3^{(n-1)}$$

Step 3: Simplify the General Term Formula (if possible)

In this case, the general term formula for the given geometric sequence cannot be simplified further: $$a_n = -\frac{1}{5} \cdot 3^{(n-1)}$$ This formula will give us the nth term of the geometric sequence.

Key Concepts

Common Ratio

When working with geometric sequences, the first step is to identify the common ratio. The common ratio is a crucial element, as it tells us how each term in the sequence relates to the previous one. For any geometric sequence, you can find the common ratio (\(r\)) by dividing any term by its preceding term. This remains constant throughout the sequence.

To illustrate, using our sequence \(-\frac{1}{5},-\frac{3}{10},-\frac{9}{20},-\frac{27}{40}, \dots\), calculate the common ratio by dividing the second term by the first: \(r = \frac{-\frac{3}{10}}{-\frac{1}{5}}\). By simplifying, we multiply \(\frac{3}{10}\) by \(\frac{5}{1}\) to get \(3\). So, the common ratio is \(3\). This constant multiplication factor helps shape the sequence, making it geometric.

General Term Formula

Once the common ratio is known, the next step is to find the general term formula for the sequence. This formula allows us to find any term in the sequence without listing all previous terms. The general term of a geometric sequence is captured by the formula: \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.

For our specific sequence where the first term \(a_1 = -\frac{1}{5}\) and the common ratio \(r = 3\), substitute these values into the formula: \(a_n = -\frac{1}{5} \cdot 3^{(n-1)}\). This formula gives us the precise position and value of each term in the sequence.

Nth Term of a Sequence

The nth term in a geometric sequence gives you the value of the term at any position in the sequence. Knowing how to calculate the nth term is crucial for solving many problems involving sequences.

Using the general term formula \(a_n = a_1 \cdot r^{(n-1)}\), simply plug in the term number you are interested in for \(n\). This is especially helpful for large sequences where listing every term manually isn't practical.

• For example, if you want to find the 5th term (\(n = 5\)) of our initial sequence, plug \(n = 5\) into the formula, \(a_5 = -\frac{1}{5} \cdot 3^{(5-1)}\), and simplify.
• This gives you the specific 5th term value, demonstrating the power of the general term formula to quickly access any term in the sequence without the need for exhaustive enumeration.