Problem 26 - Beginning and Intermediate Algebra

Question

Find a formula for the general term, \(a_{n},\) of each sequence. $$\frac{4}{5}, \frac{4}{25}, \frac{4}{125}, \frac{4}{625}, \dots$$

Step-by-Step Solution

Answer

The formula for the general term, \(a_{n}\), of the given sequence is: \[a_{n} = \frac{4}{5} \cdot \left(\frac{1}{5}\right)^{n-1}\]

Step 1: Determine if the sequence is geometric

Since all the terms are fractions and only the denominators change, we need to find out if the denominators have the same ratio. We can do this by diving each term by the term that comes before it. \[\frac{\frac{4}{25}}{\frac{4}{5}} = \frac{4}{25} \cdot \frac{5}{4} = \frac{4 \cdot 5}{4 \cdot 25} = \frac{1}{5}\] \[\frac{\frac{4}{125}}{\frac{4}{25}} = \frac{4}{125} \cdot \frac{25}{4} = \frac{4 \cdot 25}{4 \cdot 125} = \frac{1}{5}\] Since the ratio of consecutive terms is constant, the sequence is indeed geometric.

Step 2: Identify the first term and the common ratio

The first term of the sequence is given as \(a_{1}=\frac{4}{5}\), and we have determined that the common ratio, denoted by r, is \(\frac{1}{5}\).

Step 3: Write the formula for the nth term of a geometric sequence

The general formula for the nth term of a geometric sequence is: \[a_{n} = a_{1} \cdot r^{n-1}\]

Step 4: Apply the known values to the formula

We have \(a_{1}=\frac{4}{5}\) and \(r=\frac{1}{5}\). Substitute these values into the formula: \[a_{n} = \frac{4}{5} \cdot \left(\frac{1}{5}\right)^{n-1}\]

Step 5: Simplify the formula (if possible)

In this case, the formula for the nth term, \(a_{n}\), of the sequence is already simplified. So, the formula for the general term, \(a_{n}\), of the given sequence is: \[a_{n} = \frac{4}{5} \cdot \left(\frac{1}{5}\right)^{n-1}\]

Key Concepts

nth term formula

The nth term formula is a shortcut to find any term in a geometric sequence without listing all previous terms. It helps you jump straight to the term you need. In a geometric sequence, this formula looks like this:

• \( a_{n} = a_{1} \cdot r^{n-1} \)

This formula has three parts you need to know:

• \( a_{n} \) is the term you're trying to find.
• \( a_{1} \) is the first term of the sequence.
• \( r \) is the common ratio, which we determine by dividing any term by the one before it.

The exponent \( n-1 \) tells us how many times the common ratio is multiplied by itself. This is because the first term \( a_{1} \) does not require any multiplication, the second term involves multiplying by the ratio once, and so on. Always remember that using this formula requires knowing the first term and the common ratio as they are essential for correctly determining any term in the sequence.

common ratio

The common ratio is a key part of understanding geometric sequences. It helps distinguish geometric sequences from other sequence types. To find the common ratio, you take any term in the sequence and divide it by the previous term.For example, in the sequence \( \frac{4}{5}, \frac{4}{25}, \frac{4}{125}, \frac{4}{625} \), you pick any two consecutive terms:

• Divide the second term \( \frac{4}{25} \) by the first term \( \frac{4}{5} \), and you'll calculate \( \frac{1}{5} \).
• Do the same for \( \frac{4}{125} \) divided by \( \frac{4}{25} \), and you'll also get \( \frac{1}{5} \).

Repeating this process will show that the ratio is always the same, which proves the sequence is geometric. This consistency is what defines a geometric sequence, allowing every term to be generated by multiplying the previous term by the constant ratio.

sequence problems

Sequence problems often involve figuring out patterns in a series of numbers and predicting future terms. They test your ability to recognize these patterns and use formulas to find specific terms efficiently, saving you from listing each term one by one. In sequence problems, you'll often:

• Determine if a sequence is geometric or another type based on how terms change.
• Find the first term and common ratio if it's geometric.
• Use the nth term formula to find specific terms or predict further numbers in the sequence.

Understanding sequence problems is about using formulas and knowing how each part relates to others. It's like solving a puzzle where each clue relies on the last. Thus, you can grasp large sets of data and patterns without manually calculating each step. This approach is useful in various fields like finance, computer science, and even art design, where patterns play a crucial role.