Question

Write the first six terms of the sequence. (See Example 1.) \(f(n)=n^3+2\)

Short answer

The first six terms of the sequence are 3, 10, 29, 66, 127, and 218.

Step-by-step solution

Understanding 'n'

In this exercise, 'n' stands for the term number in the sequence. This means for the first term, n is 1, for the second term, n is 2, and so forth.

Apply the function to each term

Using the function \(f(n) = n^3 +2\), we will find the first six terms by replacing 'n' with 1, 2, 3, 4, 5, and 6 respectively.

Calculate the first term

For the first term, where \(n=1\), calculate \(1^3 + 2 = 3\)

Calculate the second term

For the second term, where \(n=2\), calculate \(2^3 + 2 = 10\)

Calculate the third term

For the third term, where \(n=3\), calculate \(3^3 + 2 = 29\)

Calculate the fourth term

For the fourth term, where \(n=4\), calculate \(4^3 + 2 = 66\)

Calculate the fifth term

For the fifth term, where \(n=5\), calculate \(5^3 + 2 = 127\)

Calculate the sixth term

For the sixth term, where \(n=6\), calculate \(6^3 + 2 = 218\)

Key concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a common difference between consecutive terms. In simpler terms, you add or subtract the same value every time to get from one term to the next. To find any term of an arithmetic sequence, you use the formula:

t_n = t_1 + (n-1)d

where

• t_n is the nth term you're looking for,
• t_1 is the first term in the sequence, and
• d is the common difference.

However, not all sequences are arithmetic. For example, the sequence provided in the exercise does not have a constant difference between terms; hence it is not an arithmetic sequence. Instead, it’s a type of sequence known as a cubic sequence.

Cubic Functions

A cubic function, as seen in the given exercise, is a polynomial of degree three. It has the general form:

f(x)=ax^3+bx^2+cx+d

Where

• a, b, c and d are constants,
• and 'x' is the variable.

The graph of a cubic function is a curved line that may twist and turn but will always have a single rounded peak or trough known as the inflection point. In the exercise, the cubic function is simplified to

f(n) = n^3 + 2

which means there is a cube term, no square or linear term, and a constant term. When you input different values of 'n', you get the terms of the sequence.

Mathematical Sequences

In mathematics, a sequence is an ordered list of numbers where each number is called a term. Sequences can be finite or infinite, and they can follow different patterns, such as:

• Arithmetic: with a constant difference between terms,
• Geometric: with a constant ratio between terms,
• or Cubic: where terms are based on raising the term number to the third power and possibly adding or subtracting other constants.

The sequence in our exercise is derived from a cubic function, which is a more complex form of mathematical sequences. These sequences can describe many natural phenomena and can be quite useful in solving real-world problems.

Function Notation

Function notation is a way to denote functions in mathematics. It is a shorthand that tells us which operation to perform on a set of numbers to produce another set of numbers. In function notation, we write

f(x)

which denotes a function named 'f' with 'x' as the input variable. For the provided exercise, the function is written as

f(n) = n^3 + 2

indicating that 'f' is a function depending on a variable 'n'. You replace 'n' with each term number to generate the values of the sequence. Understanding function notation is crucial for grasping the concept of functions in algebra and calculus.