Problem 94 - Algebra 2

Question

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. \(a_{n}=3 n(n+1)\)

Step-by-Step Solution

Answer

The given formula is explicit. The first five terms of the sequence are 6, 18, 36, 60, and 90.

Step 1: Determine if the Formula is Explicit or Recursive

The formula provided is \(a_{n}=3 n(n+1)\). This is an explicit formula because the nth term can be calculated directly using any positive integer 'n', without needing to know any previous terms. Hence, the sequence is explicitly defined.

Step 2: Calculate the First Five Terms

Substitute the first five positive integers (n = 1, 2, 3, 4, 5) respectively into the formula to get the first five terms of the sequence: \n\nWhen n=1, \(a_{1}=3*1*(1+1)=6\)\nWhen n=2, \(a_{2}=3*2*(2+1)=18\)\nWhen n=3, \(a_{3}=3*3*(3+1)=36\)\nWhen n=4, \(a_{4}=3*4*(4+1)=60\)\nWhen n=5, \(a_{5}=3*5*(5+1)=90\)

Key Concepts

Explicit Formula

An explicit formula allows us to directly calculate any term in a sequence without needing to know the previous terms.
This type of formula is characterized by an expression that defines the nth term as a function of n.
In other words, if you want to find a specific term in the sequence, you just substitute the desired term number into the formula. For example, consider the formula \( a_{n} = 3n(n+1) \).
This explicit formula tells us that to find the nth term, we simply substitute n into the formula, making it straightforward and quick to use.
This is especially useful when you're trying to find large sequence terms, as you don't need to compute all prior terms first. Using an explicit formula, you can:

• Identify any term directly.
• Calculate terms independently from each other.
• Avoid the accumulation of calculation errors from prior steps.

For the sequence defined by \( a_{n} = 3n(n+1) \), each term can be obtained by substituting n with 1, 2, 3, etc., as shown in the original solution.

Sequence Terms

When working with sequences, the term refers to each individual number within that sequence.
Sequence terms are usually denoted as \( a_{n} \), where n indicates the position of the term in the sequence starting from n = 1.
For example, the first five terms of the sequence generated by the formula \( a_{n} = 3n(n+1) \) can be calculated as follows:

• The 1st term: Plugging n = 1 into the formula gives \( a_{1} = 6 \).
• The 2nd term: With n = 2, we find \( a_{2} = 18 \).
• The 3rd term: For n = 3, \( a_{3} = 36 \).
• The 4th term: When n = 4, \( a_{4} = 60 \).
• The 5th term: Finally, n = 5 gives \( a_{5} = 90 \).

Each sequence term follows directly from the explicit formula, providing a systematic approach to sequence calculations.
This patternmaking makes it easier to understand and predict future terms.
Through this method, even complex sequences become manageable.

Mathematical Sequences

Mathematical sequences are lists of numbers arranged in a specific order by following a particular rule or formula.
There are many types of sequences, including arithmetic, geometric, and special formulas, such as the one given in the exercise.
Understanding the properties of sequences helps in exploring patterns and solving complex mathematical problems effortlessly. Sequences are defined by rules:

• Arithmetic sequences progress by adding or subtracting a constant.
• Geometric sequences involve multiplying by a constant.
• Explicit formulas, like our example, provide a direct calculation method for any term.

Recognizing the sequence type is crucial for choosing the right approach to finding terms or solving problems linked to sequences.
In mathematics, sequences provide a framework for modeling real-world scenarios, predicting outcomes, or finding patterns.
The sequence from the exercise, using an explicit formula, is easily understood and serves as a basis for further exploration into more complex structures.