Certainly! Let's examine the given sequence of numbers:
[tex]\[ 1, \quad 8, \quad 27, \quad 64, \quad \ldots \][/tex]
First, recognize that these numbers represent a specific pattern.
1. The first number in the sequence is [tex]\(1\)[/tex].
2. The second number in the sequence is [tex]\(8\)[/tex].
3. The third number in the sequence is [tex]\(27\)[/tex].
4. The fourth number in the sequence is [tex]\(64\)[/tex].
To identify the pattern, let's look at each number:
- [tex]\(1\)[/tex] can be written as [tex]\(1^3\)[/tex].
- [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex].
- [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex].
- [tex]\(64\)[/tex] can be written as [tex]\(4^3\)[/tex].
Clearly, each number in the sequence is the cube of an integer, starting from [tex]\(1\)[/tex].
So, the [tex]\(n\)[/tex]-th term of this sequence appears to be [tex]\(n^3\)[/tex].
For example:
- The 1st term is [tex]\(1^3 = 1\)[/tex].
- The 2nd term is [tex]\(2^3 = 8\)[/tex].
- The 3rd term is [tex]\(3^3 = 27\)[/tex].
- The 4th term is [tex]\(4^3 = 64\)[/tex].
To find the next term in the sequence, we need the cube of the next integer, which is [tex]\(5\)[/tex].
Thus, the next term is:
[tex]\[ 5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125 \][/tex]
Therefore, the next term in the sequence is:
[tex]\[ 125 \][/tex]
