Problem 2 - Essential Calculus

Question

Write the first five terms of the sequence. \(a_{n}=\frac{3^{n}}{n !}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are \(3, 4.5, 4.5, 6.75, 6.75\)

Step 1: apply the sequence formula for \(n = 1\)

Substitute \(n=1\) into the sequence formula: \(a_{1}=\frac{3^{1}}{1!}=3\)

Step 2: apply the sequence formula for \(n = 2\)

Substitute \(n=2\) into the sequence formula: \(a_{2}=\frac{3^{2}}{2!}=4.5\)

Step 3: apply the sequence formula for \(n = 3\)

Substitute \(n=3\) into the sequence formula: \(a_{3}=\frac{3^{3}}{3!}=4.5\)

Step 4: apply the sequence formula for \(n = 4\)

Substitute \(n=4\) into the sequence formula: \(a_{4}=\frac{3^{4}}{4!}=6.75\)

Step 5: apply the sequence formula for \(n = 5\)

Substitute \(n=5\) into the sequence formula: \(a_{5}=\frac{3^{5}}{5!}=6.75\)

Key Concepts

Factorials

Factorials are a key concept in mathematics, especially when dealing with sequences and series. The factorial of a number, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to that number. For example, the factorial of 3 is calculated as follows:

• 3! = 3 × 2 × 1 = 6

Factorials are typically used in permutations, combinations, and finding successive elements in sequences. In the provided sequence formula, factorials help determine the denominator. As the number on the bottom grows quickly with higher values of n, it affects the outcome of each term in the sequence because the growth of the factorial reduces the value of the fraction. This is an essential point for understanding how sequences work. You'll notice that despite the exponential growth in the numerator, the factorial's rapid increase in the denominator can stabilize or even decrease the sequence values as n becomes larger.

Mathematical Notation

Mathematical notation is a system of symbols used to express mathematical concepts. It allows mathematicians and students to communicate more efficiently about complex ideas. In sequences, you often see notation such as

• Variables with subscripts like \(a_n\) indicating a sequence term dependent on n.
• Exponents such as \(3^n\) for repeated multiplication of a base number.
• Factorials denoted by the exclamation mark \(!\).

Learning to read and write this mathematical language is crucial as it allows for easy manipulation and understanding of complex formulas. In our sequence formula \(a_n = \frac{3^n}{n!}\), the symbols all play a role. Understanding each part will help you see how changes in n influence the outcome, by affecting the exponential and factorial aspects of the term.

Algebra

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. It's fundamental when working with sequences because it provides a system for solving equations and understanding how changing one variable affects the whole expression. In the sequence given, algebraic principles allow us to replace n with different numbers to find how the sequence progresses. This involves substitution and the simplification of expressions — vital algebraic skills. For example, to find \(a_3\), you substitute \(n = 3\) in the formula:

• First calculate \(3^3 = 27\).
• Then compute \(3! = 3 \times 2 \times 1 = 6\).
• Finally, simplify \(\frac{27}{6} = 4.5\).

Understanding algebra helps to manipulate and simplify expressions like these, making it easier to calculate and understand how each term in the sequence is generated. By practicing how to use algebraic techniques with sequences, you'll gain a deeper understanding of both topics.