Problem 28 - Algebra 2

Question

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=-4 n^{2}-2 $$

Step-by-Step Solution

Answer

The formula is an explicit formula. The first five terms of the sequence are -6, -18, -38, -66, -102.

Step 1: Identify the Formula Type

Looking at the given formula \(a_{n}=-4 n^{2}-2\), it is observed that each term is defined independently as a function of n. Hence, the formula is an explicit formula.

Step 2: Compute the First Five Terms

Replace n with 1, 2, 3, 4, and 5 in the equation to find the first five terms respectively. This results in: \(a_{1}=-4(1)^{2}-2=-6\), \(a_{2}=-4(2)^{2}-2=-18\), \(a_{3}=-4(3)^{2}-2=-38\), \(a_{4}=-4(4)^{2}-2=-66\), \(a_{5}=-4(5)^{2}-2=-102\).

Key Concepts

Recursive Formula

Understanding a recursive formula is crucial for sequence analysis. Unlike an explicit formula, a recursive formula defines each term based on the preceding term or terms. This means that to find any term in a sequence, you often need to know the term before it. For instance, consider a simple arithmetic sequence where each term is the previous term plus a constant. A recursive formula for such a sequence might look like this:

• \( a_n = a_{n-1} + d \)

Here, \(d\) represents the common difference between consecutive terms. This formula is useful when you need to build a sequence step-by-step, but it requires the knowledge of one or more previous terms. Recursive formulas are particularly effective for sequences that are defined by their progression, such as Fibonacci sequences. However, they are not always efficient for computing distant terms without a computer program, as you'll need to compute every previous term to arrive at the desired position in the sequence.

Quadratic Sequence

A quadratic sequence is a sequence of numbers where the second differences are constant. This means that when you take the differences between consecutive terms twice, the result is the same. Quadratic sequences are often described by a quadratic formula, which involves a square term, such as \( n^2 \). The typical form of a quadratic sequence formula looks something like:

• \( a_n = an^2 + bn + c \)

Where \(a, b,\) and \(c\) are constants. Notice how similar this is to the standard quadratic equation you might have seen: \(ax^2 + bx + c = 0\). For example, in the formula \( a_{n}=-4n^{2}-2 \), the sequence will yield quadratic behavior due to the \(-4n^2\) term. Quadratic sequences can model various real-world phenomena like projectile motion, where an object's height at time \(n\) can be defined by such equations.

Sequence Terms

In any sequence, understanding what sequence terms are is foundational. Sequence terms are essentially the elements of a sequence—these are the values generated based on the given formula. Sequences can have infinitely many terms or be finite, depending on their definition. To generate terms in a sequence, substitute values of \(n\) into the sequence formula. For example, substituting \(n = 1, 2, 3, \ldots\) in our formula \(a_{n}=-4n^{2}-2\) gives:

• \(a_1 = -6\)
• \(a_2 = -18\)
• \(a_3 = -38\)
• \(a_4 = -66\)
• \(a_5 = -102\)

Each term here is calculated independently by plugging \(n\) into the given formula, which, based on its form, is known as the explicit formula. Each of these terms tells us the position-specific value in the quadratic sequence.

Mathematical Patterns

Mathematical patterns are fundamental to understanding sequences and series. Patterns allow us to predict, generalize, and establish rules within a sequence. Recognizing these patterns can often simplify complex mathematical problems and make sense of numerical behaviors and trends. There are different types of patterns: arithmetic, geometric, and quadratic, among others. An arithmetic pattern involves a constant difference between terms. In contrast, a geometric pattern involves a constant ratio. A quadratic pattern, like the one in our example, involves a constant second difference—indicative of the quadratic term consolidation. Detecting these patterns generally involves:

• Identifying constant differences or ratios.
• Observing how the terms increase or decrease in a sequence.
• Using formulas to extrapolate or predict further terms.

Patterns play an integral role in forming the basis for recursive and explicit formulas, helping mathematicians and students alike to extrapolate and understand complex information from simple sequences.