Question

Are the sequences geometric? For those that are, give a formula for the \(n^{\text {th }}\) term. $$ 4,20,100,500, \ldots $$

Short answer

Answer: The formula for the nth term of the given geometric sequence is $$a_n = 4 \cdot 5^{n-1}$$.

Step-by-step solution

Identify the terms in the sequence

The given sequence has terms: $$ 4, 20, 100, 500, \ldots $$

Determine if there is a common ratio between terms

Divide each term by the previous term: $$ \frac{20}{4}=5, \quad \frac{100}{20}=5, \quad \frac{500}{100}=5 $$ As each consecutive term divided by its previous term results in a constant value, we can conclude that the sequence is geometric, with a common ratio of 5.

Create a formula for the nth term of the sequence

Now that we know the sequence is geometric with a common ratio of 5, we can create the formula for the nth term. A geometric sequence can be written using the following formula: $$ a_n = a_1 \cdot r^{n-1} $$ In this case, we have the first term, \(a_1=4\), and the common ratio \(r=5\). So the formula for the nth term is: $$ a_n = 4 \cdot 5^{n-1} $$ Now we've found the formula for the nth term of the sequence, so we're done.

Key concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply each term to get the next term. This makes geometric sequences unique as they have a consistent multiplicative pattern that defines them.
To find the common ratio, you take any term in the sequence and divide it by the previous term. In our example sequence:

• The second term is 20, and the first term is 4. Dividing them gives you \( \frac{20}{4} = 5 \).
• The third term is 100, and the second term is 20. Here, \( \frac{100}{20} = 5 \).
• The fourth term is 500, and the third term is 100, with \( \frac{500}{100} = 5 \).

In all cases, we get the common ratio of 5, confirming that the sequence shows a consistent pattern of multiplication by 5. Knowing this ratio is crucial because it helps in deducing other characteristics of the sequence, including finding other terms fast using a simple formula.

Nth Term Formula

The nth term formula is a fundamental expression in understanding geometric sequences. It allows you to find any term in the sequence without having to list all preceding terms.
The formula for a geometric sequence is expressed as: \[ a_n = a_1 \, \cdot \, r^{n-1} \] where:

• \( a_n \) is the term you are trying to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio of the sequence.
• \( n \) is the position of the term in the sequence.

For our specific sequence, with \( a_1 = 4 \) and \( r = 5 \), the nth term formula becomes: \[ a_n = 4 \, \cdot \, 5^{n-1} \] This formula allows you to easily calculate any term in the pattern, providing a quick way to understand the sequence's progression without manually listing all the terms.

Sequence Analysis

Sequence analysis involves examining the properties and characteristics of a sequence to understand it thoroughly. For geometric sequences, this includes identifying the terms, the common ratio, and deriving the nth term formula.
Let's break it down:

• Start by identifying terms: Before anything, you need to list at least a few terms from the sequence. For example, we have 4, 20, 100, and 500.
• Confirm the sequence type: Calculate the common ratios to ensure every term is consistently multiplied by the same value.
• Use the results: Utilize these findings to derive broader properties and the nth term formula for extended sequence analysis.

By following these steps, one can confirm whether a sequence is geometric and then fully explore its characteristics. Such comprehensive analysis not only helps answer immediate problems but also builds a deeper understanding for more complex mathematical scenarios.