Problem 5 - Beginning and Intermediate Algebra

Question

Fill in the blanks to complete the terms of each geometric sequence. \(\frac{1}{3},-\frac{1}{9}, \frac{1}{27}\) _____,_____,_____.

Step-by-Step Solution

Answer

The next three terms are \(-\frac{1}{81}, \frac{1}{243}, -\frac{1}{729}\).

Step 1: Identify the common ratio

To find the common ratio in a geometric sequence, divide the second term by the first term or the third term by the second term. Here, calculate the common ratio by dividing the second term \(-\frac{1}{9}\) by the first term \(\frac{1}{3}\): i = \frac{-\frac{1}{9}}{\frac{1}{3}} = - \frac{1}{9} \times \frac{3}{1} = - \frac{3}{9} = - \frac{1}{3}.Therefore, the common ratio is \(-\frac{1}{3}\).

Step 2: Multiply to find the next term

To find the next term in the sequence, multiply the last known term (third term \(\frac{1}{27}\)) by the common ratio \(-\frac{1}{3}\). i = \frac{1}{27} \times -\frac{1}{3} = - \frac{1}{81}.

Step 3: Continue multiplying to find subsequent terms

Use the common ratio to find the remaining terms by continuing to multiply the last known term by \(-\frac{1}{3}\). For the fifth term: i = - \frac{1}{81} \times - \frac{1}{3} = \frac{1}{243}.And for the sixth term: i = \frac{1}{243} \times - \frac{1}{3} = - \frac{1}{729}.

Key Concepts

Common Ratio

In a geometric sequence, one of the most important concepts is the common ratio. This is the number you multiply by to get from one term to the next.
It's what makes the sequence 'geometric'.
To find the common ratio, divide the second term by the first term or any term by the term that comes before it.
For example, in the sequence \(\frac{1}{3}, -\frac{1}{9}, \frac{1}{27}\), you find the common ratio by dividing \(-\frac{1}{9}\) by \(\frac{1}{3}\): \(\frac{-\frac{1}{9}}{\frac{1}{3}} = - \frac{1}{3}\). This tells you that each term is \(\frac{1}{3}\) of the previous term and it's negative.
Remember, the common ratio can be positive or negative. In our example, it's negative which makes the sequence alternates between positive and negative numbers.
Once you've found the common ratio, it becomes a simple process of multiplication to find any term in the sequence.

Multiplication in Sequences

In geometric sequences, multiplication is the key to figuring out each successive term. Once you have the common ratio, you can find future terms by repeatedly multiplying by this ratio.
Let's revisit the sequence \(\frac{1}{3}, -\frac{1}{9}, \frac{1}{27}\). We found the common ratio to be \(-\frac{1}{3}\). To find the next term, we take the last known term and multiply it by \(-\frac{1}{3}\): \(\frac{1}{27} \times -\frac{1}{3} = - \frac{1}{81}\).
This keeps the sequence going. For the fifth term, multiply \(-\frac{1}{81} \times -\frac{1}{3}\): \(\frac{1}{243}\).
Notice how you only need the most recent term and the common ratio to keep going. Multiplication makes it straightforward to continue the sequence, whether you're moving forward or backwards.

Negative Fractions

Dealing with negative fractions in geometric sequences may seem tricky at first, but it's manageable once you understand the basic rules.
When a fraction is negative, it changes the sign of your product each time you multiply. For example, in our sequence \(\frac{1}{3}, -\frac{1}{9}, \frac{1}{27}\): the first term is positive, the second term becomes negative \(-\frac{1}{9}\), and the third term returns to positive \(\frac{1}{27}\). Notice the alternating signs.
If the negative sign is part of your common ratio, as in \(-\frac{1}{3}\), each multiplication will flip the sign of your result.
Understanding this flip-flopping of signs is crucial in geometric sequences with negative fractions.