Question

List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{3^{n}}{n}\right\\}\)

Short answer

3, 4.5, 9, 20.25, 48.6

Step-by-step solution

Identify the formula for the sequence

The given sequence is described by the formula \( a_{n} = \frac{3^{n}}{n} \). This formula will be used to calculate the first five terms.

Calculate the first term

To find the first term, substitute \( n = 1 \) into the formula: \[ a_{1} = \frac{3^{1}}{1} = 3 \]

Calculate the second term

To find the second term, substitute \( n = 2 \) into the formula: \[ a_{2} = \frac{3^{2}}{2} = \frac{9}{2} = 4.5 \]

Calculate the third term

To find the third term, substitute \( n = 3 \) into the formula: \[ a_{3} = \frac{3^{3}}{3} = \frac{27}{3} = 9 \]

Calculate the fourth term

To find the fourth term, substitute \( n = 4 \) into the formula: \[ a_{4} = \frac{3^{4}}{4} = \frac{81}{4} = 20.25 \]

Calculate the fifth term

To find the fifth term, substitute \( n = 5 \) into the formula: \[ a_{5} = \frac{3^{5}}{5} = \frac{243}{5} = 48.6 \]

Key concepts

sequence terms

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term.
Terms are usually represented by symbols like \(a_n\), where \(n\) indicates the position of the term in the sequence.
For example, the first term is often written as \(a_1\), the second term as \(a_2\), and so on.
In the exercise provided, the formula \(a_n = \frac{3^n}{n}\) is used to generate the terms of the sequence.

By substituting different values of \(n\), such as 1, 2, 3, etc., into the formula, we can find the corresponding terms:

• For \(n = 1\), the first term is 3
• For \(n = 2\), the second term is 4.5
• For \(n = 3\), the third term is 9
• For \(n = 4\), the fourth term is 20.25
• For \(n = 5\), the fifth term is 48.6

Understanding how to calculate sequence terms using a given formula is crucial for solving many mathematical problems.

arithmetic sequences

While the sequence in the exercise is not an arithmetic sequence, understanding arithmetic sequences is still important.
In an arithmetic sequence, the difference between consecutive terms is always the same.
This difference is called the common difference and is usually denoted by \(d\).
The general formula for the \(n\)-th term of an arithmetic sequence is: \[ a_n = a_1 + (n-1) \times d\]
where \(a_1\) is the first term and \(d\) is the common difference.

• For example, if the first term is 2 and the common difference is 3, the sequence would be 2, 5, 8, 11, 14, and so on.

This type of sequence appears frequently in various areas of algebra and is a foundational concept.

algebra formulas

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations.
These formulas form the backbone of many mathematical problems, including those involving sequences.
For example, the sequence formula \(a_n = \frac{3^n}{n}\) in the exercise is an algebraic expression.
By substituting different values of \(n\), we can calculate specific terms in the sequence.

• Another common algebra formula is the quadratic formula, which is used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The solution is given by:
  \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Mastering these formulas allows students to solve a wide range of mathematical problems with confidence.
Remember, practice is key when it comes to becoming comfortable with algebra and sequence calculations.