Problem 9 - Algebra and Trigonometry

Question

Find the first four terms and the 100th term of the sequence. \(a_{n}=n^{n}\)

Step-by-Step Solution

Answer

The first four terms are 1, 4, 27, 256. The 100th term is 100^100.

Step 1: Identify the Pattern for the Sequence

The sequence is defined by the formula \(a_{n} = n^n\). Each term in the sequence is generated by raising \(n\) to the power of \(n\). We will find the first four terms by substituting \(n = 1\), \(n = 2\), \(n = 3\), and \(n = 4\) into the formula.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula: \(a_{1} = 1^{1} = 1\).

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \(a_{2} = 2^{2} = 4\).

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula: \(a_{3} = 3^{3} = 27\).

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \(a_{4} = 4^{4} = 256\).

Step 6: Find the 100th Term

Substitute \(n = 100\) into the formula: \(a_{100} = 100^{100}\). This represents a very large number with 100 trailing zeros.

Key Concepts

Exponents

Exponents are a mathematical tool that shows you how many times to multiply a number by itself. For example, in the expression \( n^n \), \( n \) is the base, and the exponent is \( n \) itself. This tells us to multiply \( n \) by itself \( n \) times. Exponents simplify expressions and make large numbers easier to work with.When you are working with exponents, try to memorize some basic powers. For instance, \( 2^3 = 8 \), which means multiplying 2 three times: \( 2 \times 2 \times 2 \). Knowing these can speed up your calculations. Understanding exponents is crucial for understanding patterns in sequences and solving problems efficiently.

Term Calculation

To find a specific term in an exponential sequence, you need to substitute the term number into the given formula. Take the sequence formula \( a_n = n^n \), for example. Here, the term number \( n \) is used as both the base and the exponent.Let's calculate with this formula:

• First Term: Set \( n = 1 \), so \( a_1 = 1^1 = 1 \).
• Second Term: Set \( n = 2 \), so \( a_2 = 2^2 = 4 \).
• Third Term: Set \( n = 3 \), so \( a_3 = 3^3 = 27 \).
• Fourth Term: Set \( n = 4 \), so \( a_4 = 4^4 = 256 \).

To calculate the 100th term, you would plug in \( n = 100 \), resulting in \( a_{100} = 100^{100} \), which, as you can imagine, is an immense number. Breaking down steps like these helps in accurately finding terms without error.

Sequence Patterns

Mathematical sequences rely on patterns to understand their nature. In an exponential sequence like \( a_n = n^n \), each term is generated by a predictable pattern, multiplying a number by itself \( n \) times. This causes each term to grow rapidly.Recognizing these patterns helps solve sequence-related problems more swiftly.

• An exponential growth pattern results in increasingly larger numbers.
• Terms are not added or subtracted but instead, each term grows relative to itself.

Patterns are crucial because they allow for predictions about the sequence's behavior, such as calculating far-reaching terms without computing each one step by step. Understanding these patterns not only simplifies calculations but also provides deeper insights into how exponential sequences behave.