Problem 22 - Algebra and Trigonometry

Question

Find the first five terms of the sequence and determine if 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; terms are 1, 4, 27, 256, 3125.

Step 1: Find the first term

To find the first term of the sequence, substitute \( n = 1 \) into the formula \( a_n = n^n \). So, \( a_1 = 1^1 = 1 \).

Step 2: Find the second term

To find the second term, substitute \( n = 2 \) into the formula \( a_n = n^n \). So, \( a_2 = 2^2 = 4 \).

Step 3: Find the third term

To find the third term, substitute \( n = 3 \) into the formula \( a_n = n^n \). So, \( a_3 = 3^3 = 27 \).

Step 4: Find the fourth term

To find the fourth term, substitute \( n = 4 \) into the formula \( a_n = n^n \). So, \( a_4 = 4^4 = 256 \).

Step 5: Find the fifth term

To find the fifth term, substitute \( n = 5 \) into the formula \( a_n = n^n \). So, \( a_5 = 5^5 = 3125 \).

Step 6: Determine if sequence is geometric

A sequence is geometric if the ratio of consecutive terms is constant. Calculate the ratios: \( \frac{a_2}{a_1} = \frac{4}{1} = 4 \), \( \frac{a_3}{a_2} = \frac{27}{4} \), \( \frac{a_4}{a_3} = \frac{256}{27} \), and \( \frac{a_5}{a_4} = \frac{3125}{256} \). These are not equal, so the sequence is not geometric.

Key Concepts

Sequence and Series

A **sequence** is an ordered list of numbers where each number is called a term. Sequences can be based on many different rules. In a mathematical sequence, there is often a rule that determines how each term is related to the previous one.
In the context of mathematics, a **series** is the sum of the terms of a sequence.

To better grasp these concepts, observe how sequences are structured:

• A given sequence of numbers could be: 2, 4, 6, 8, ..., where each term is increased by adding 2 to the previous one.
• Another sequence could be 1, 4, 9, 16, ..., where each term is a perfect square of consecutive integers (1, 2, 3, 4, ...).

The rules defining sequences can be different, leading to various types like arithmetic and geometric sequences. Understanding these differences is vital for solving problems involving them.

Common Ratio

The term **common ratio** appears when discussing a specific type of sequence known as a geometric sequence. In a geometric sequence, each term is formed by multiplying the previous term by a fixed, non-zero number called the common ratio. Let's delve deeper:

• Imagine a sequence: 3, 6, 12, 24, ..., where each number is multiplied by 2, so the common ratio is 2.
• In a geometric sequence with a common ratio of 3, starting from 5, you'd have: 5, 15, 45, 135, ..., continuing indefinitely.

To determine if a sequence is geometric, the ratio of any term to its preceding term must remain constant throughout. If this condition is not met, the sequence is not geometric, as was the case in our earlier exercise where ratios varied for each pair of terms.

General Term of a Sequence

The **general term** of a sequence is an expression that allows us to find any term in the sequence without computing all the preceding terms. This term is particularly useful as it provides a formula based on position within the sequence.For geometric sequences, the general term is expressed as\[ a_n = a \cdot r^{n-1} \]where:

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

This formula is exceptionally valuable as it means any term can be calculated directly by knowing the first term and the common ratio.

In our example, however, the sequence was not geometric, implying that an alternative approach is needed since the sequence could not be expressed in the standard geometric form.