Problem 26 - Precalculus Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}\). $$a_{n}=n^{n}$$

Step-by-Step Solution

Answer

The sequence is not geometric.

Step 1: Find the First Five Terms

Calculate the first five terms of the sequence given by the function \( a_{n} = n^n \). This means evaluating the function for \( n = 1 \) to \( n = 5 \): - \( a_1 = 1^1 = 1 \)- \( a_2 = 2^2 = 4 \)- \( a_3 = 3^3 = 27 \)- \( a_4 = 4^4 = 256 \)- \( a_5 = 5^5 = 3125 \)Thus, the first five terms are 1, 4, 27, 256, and 3125.

Step 2: Determine if the Sequence is Geometric

A sequence is geometric if the ratio between consecutive terms is constant. Let's find the ratios:\( \frac{a_2}{a_1} = \frac{4}{1} = 4 \)\( \frac{a_3}{a_2} = \frac{27}{4} \approx 6.75 \)\( \frac{a_4}{a_3} = \frac{256}{27} \approx 9.48 \)\( \frac{a_5}{a_4} = \frac{3125}{256} \approx 12.21 \)The ratios are not constant, so the sequence is not geometric.

Step 3: Conclusion

Since the ratios between consecutive terms are not equal, the sequence is not geometric. Therefore, there is no common ratio or formula of the form \( a_{n} = a r^{n-1} \) for this sequence.

Key Concepts

Sequence Terms

A sequence is an ordered list of numbers that usually follow a specific pattern. In this particular exercise, the sequence is defined by the function \(a_{n} = n^{n}\). This means that to find any term in the sequence, you simply need to raise the term number \(n\) to the power of itself. Here’s how you calculate the **first five terms**:

• For \(n = 1\), the term is \(a_1 = 1^1 = 1\).
• For \(n = 2\), the term is \(a_2 = 2^2 = 4\).
• For \(n = 3\), the term is \(a_3 = 3^3 = 27\).
• For \(n = 4\), the term is \(a_4 = 4^4 = 256\).
• For \(n = 5\), the term is \(a_5 = 5^5 = 3125\).

Thus, the sequence starts with the numbers: 1, 4, 27, 256, and 3125. Each term highlights a unique property of exponential growth, as the base and exponent are the same. Unlike arithmetic sequences, terms here don’t follow a simple difference rule.

Common Ratio

In a geometric sequence, the **common ratio** is a constant number that each term is multiplied by to get the next term. You might identify a geometric sequence by calculating the ratio of consecutive terms. These ratios should always yield the same number for the sequence to be considered geometric. Let's examine the ratios in our sequence:

• \(\frac{a_2}{a_1} = \frac{4}{1} = 4\)
• \(\frac{a_3}{a_2} = \frac{27}{4} \approx 6.75\)
• \(\frac{a_4}{a_3} = \frac{256}{27} \approx 9.48\)
• \(\frac{a_5}{a_4} = \frac{3125}{256} \approx 12.21\)

As these ratios are not equal, this sequence does not have a common ratio and is not geometric. In geometric sequences, consistency of this ratio is key to identifying pattern regularity, enabling easy prediction and term calculation.

Nth Term Formula

The **nth term formula** for a geometric sequence takes the form \(a_{n} = a \, r^{n-1}\), where:

• \(a\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number.

This formula allows you to find any term in a geometric sequence if you know the first term and the common ratio. However, since the sequence in our exercise \(a_n = n^n\) is not geometric, we do not use this formula. The lack of a consistent common ratio negates applying the formula, stressing the importance of recognizing the sequence type before choosing the correct formula for calculations.