Question

List the first five terms of each sequence. \(\left\\{b_{n}\right\\}=\left\\{\frac{n}{e^{n}}\right\\}\)

Short answer

\(b_1 = \frac{1}{e}\), \(b_2 = \frac{2}{e^2}\), \(b_3 = \frac{3}{e^3}\), \(b_4 = \frac{4}{e^4}\), \(b_5 = \frac{5}{e^5}\).

Step-by-step solution

- Understand the given sequence

The sequence given is \(\b_n = \frac{n}{e^n}\). To find the first five terms, substitute the values of \(n\) starting from 1 to 5.

- Substitute n = 1 into the sequence

For \(n = 1\), \(b_1 = \frac{1}{e^1} = \frac{1}{e}\).

- Substitute n = 2 into the sequence

For \(n = 2\), \(b_2 = \frac{2}{e^2} = \frac{2}{e^2}\).

- Substitute n = 3 into the sequence

For \(n = 3\), \(b_3 = \frac{3}{e^3} = \frac{3}{e^3}\).

- Substitute n = 4 into the sequence

For \(n = 4\), \(b_4 = \frac{4}{e^4} = \frac{4}{e^4}\).

- Substitute n = 5 into the sequence

For \(n = 5\), \(b_5 = \frac{5}{e^5} = \frac{5}{e^5}\).

Key concepts

exponential function

An exponential function is a mathematical expression in which a variable appears in the exponent. It is written in the form of \(a^x\), where \(a\) is the base and \(x\) is the exponent. In the context of sequences, an exponential function helps to explain how terms grow or shrink rapidly based on their position in the sequence.

For example, in the sequence \(b_n = \frac{n}{e^n}\), the term \(e^{-n}\) indicates an exponential decay because \(e\) (Euler's number, approximately equal to 2.718) raised to the power of \(-n\) quickly becomes smaller as \(n\) increases.

Understanding exponential functions allows you to grasp why terms in some sequences decrease incredibly fast. That helps you predict the behavior of later terms based on early ones.

substitution

Substitution is a critical method in sequence calculation. It involves replacing variables with specific values to calculate terms.

For sequences, you often substitute the term number (like 1, 2, 3) into the formula to find the value of that particular term.

Let's illustrate substitution using the sequence: \(b_n = \frac{n}{e^n}\).
To calculate the first term, \b_1\, replace \(n\) with 1: \(b_1 = \frac{1}{e^1}\).
Repeat for the second term: \(b_2 = \frac{2}{e^2}\).
The process continues similarly:
1. \(b_3 = \frac{3}{e^3}\),
2. \(b_4 = \frac{4}{e^4}\),
3. \(b_5 = \frac{5}{e^5}\).

sequence calculation

Sequence calculation is the process of determining specific terms within a sequence based on a given formula. This involves using math operations such as addition, multiplication, and substitution.

For the sequence \(b_n = \frac{n}{e^n}\), the goal is to find the values of \(b_1, b_2, b_3, b_4,\) and \(b_5\).

Here’s a step-by-step guide:

• Start with the formula: \(b_n = \frac{n}{e^n}\).
• Substitute \(n\) with 1: \(b_1 = \frac{1}{e^1}\).
• Substitute \(n\) with 2: \(b_2 = \frac{2}{e^2}\).
• Continue with \(n\) values 3, 4, and 5 to find \(b_3, b_4,\) and \(b_5\).

The calculations allow you to express terms explicitly, shedding light on their individual behaviors and patterns.

mathematical sequences

Mathematical sequences are ordered lists of numbers following specific rules or formulas. Each number in the list is called a term. For example, in the sequence \(b_n = \frac{n}{e^n}\), each term is derived by plugging in successive values of \(n\).

Sequences can be categorized in many ways—arithmetic, geometric, or based on more complex rules like the one using the exponential function in our example.

Understanding sequences involves:

• Recognizing patterns or rules that define each term.
• Using formulas to compute specific terms.
• Analyzing the relationship between terms.

Mastery of sequences aids in problem-solving and in various applications within different branches of mathematics and science.