Problem 35 - Algebra 2

Question

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 1,8,27,64,125, \dots $$

Step-by-Step Solution

Answer

The next two terms in the sequence are 216 and 343. The explicit formula for the nth term is \(a_n = n^3\). There's no simple recursive formula for this sequence.

Step 1: Identify the pattern and calculate the next terms

The sequence \(1,8,27,64,125, \dots \) displays a pattern where each term is a cube of a natural number. The cubic root of 1 is 1, the cubic root of 8 is 2, of 27 is 3, of 64 is 4, and of 125 is 5. So, it seems that term \(n\) in the sequence is equivalent to \(n^3\). Thus, the next two terms should be the cube of 6 and the cube of 7, which are 216 and 343 respectively.

Step 2: Identify the explicit formula

Since the term number's cube is the value of that term in the sequence, the explicit formula for this sequence is \(a_n = n^3\). Here, \(a_n\) represents the nth term.

Step 3: Identify the recursive formula

A recursive formula defines the terms of a sequence using previous terms. However, for this specific sequence it's not a simple relationship and it's much easier to use the explicit formula \(a_n = n^3\). The recursive formula would be more complex and not straightforward, hence it's not advisable to use the recursive formula in this case.

Key Concepts

Explicit Formulas

In sequences, we often want to find an easy way to describe each term mathematically without listing them. That's where the concept of an explicit formula comes in handy. An explicit formula provides a direct way to find the \(n\)th term of a sequence using the term number itself. It requires no knowledge of previous terms, making it very efficient.

For the sequence \(1, 8, 27, 64, 125, \dots\), each term is the cube of a natural number. This means for any term number \(n\), the term value is \(n^3\). Hence, the explicit formula here is:

\[ a_n = n^3 \]

• a_n is the value of the \(n\)th term.
• n is the position of the term in the sequence.

With this formula, you can easily compute any term in the sequence without having to find all the prior ones.

Recursive Formulas

Unlike explicit formulas, recursive formulas define each term of a sequence based on one or more previous terms. While powerful, recursive formulas can sometimes be less convenient because modeling each term requires understanding the terms before it.

In our given sequence, where each term is a cube, using a recursive formula becomes complex. This is because each term isn't straightforwardly derived from the last, without heavy algebraic manipulations.

Mathematically, a recursive formula tends to look like:

• a₁: The first term is provided.
• a_n: Expressed in terms of one or more previous terms like \(a_{n-1}\), \(a_{n-2}\), etc.

In this sequence, explicitly knowing that \(a_n = n^3\) simplifies the process. Although recursive formulas are useful, for this sequence, an explicit formula is more efficient.

Patterns in Sequences

Recognizing patterns in sequences is the backbone of finding formulas, both explicit and recursive. Patterns help us determine the relationships between consecutive terms and lay down the mathematical rules that represent these relationships.

For the sequence \(1, 8, 27, 64, 125, \dots\), the pattern is clear when you recognize each term as the cube of its position in the sequence:

• 1: \(1^3\)
• 8: \(2^3\)
• 27: \(3^3\)
• 64: \(4^3\)
• 125: \(5^3\)

Identifying this pattern is essential, as it allows us to quickly derive the explicit formula \(a_n = n^3\).

When faced with unknown sequences, looking for arithmetic patterns (addition or subtraction), geometric patterns (multiplication or division), or more complex ones like cubic or quadratic, can guide you in finding formulas efficiently and accurately.