Problem 66 - Precalculus with Limits

Question

In Exercises 63-66, write the first five terms of the sequence defined recursively. \( a_1 = 32, a_{k + 1} = \dfrac{1}{2} a_k \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 32, 16, 8, 4, 2.

Step 1: Identify the Initial Term

The initial term of the sequence is defined as \( a_1 = 32 \).

Step 2: Apply the Recursive Formula for \( a_2 \)

According to the recursive formula \( a_{k + 1} = \dfrac{1}{2} a_k \), the second term in the sequence is \( a_2 = \dfrac{1}{2} a_1 = \dfrac{1}{2} * 32 = 16 \).

Step 3: Apply the Recursive Formula for \( a_3 \)

Applying the recursive formula again for the third term, we get \( a_3 = \dfrac{1}{2} a_2 = \dfrac{1}{2} * 16 = 8 \).

Step 4: Apply the Recursive Formula for \( a_4 \)

Using the recursive formula for the fourth term, we get \( a_4 = \dfrac{1}{2} a_3 = \dfrac{1}{2} * 8 = 4 \).

Step 5: Apply the Recursive Formula for \( a_5 \)

Finally, applying the recursive formula for the fifth term, we get \( a_5 = \dfrac{1}{2} a_4 = \dfrac{1}{2} * 4 = 2 \).

Key Concepts

Mathematical Sequences

A mathematical sequence is an ordered list of numbers, where the order and the actual numbers follow a specific rule or set of rules. Each number in the sequence is referred to as a 'term.' Sequences can be either finite, having a definite number of terms, or infinite, continuing indefinitely. In the sequence given in the exercise, the order of terms is not random but follows a defined pattern.

• The first term is explicitly given as 32.
• Each subsequent term is half of the previous term.

This type of sequence, where each term depends on one or more previous terms, introduces us to the concept of recursion.

Recursion in Mathematics

Recursion is a fundamental concept in mathematics and computer science, where a function or process calls itself with the aim to solve a problem by breaking it down into simpler subproblems. In terms of mathematical sequences, recursion allows us to define a sequence where each term is generated based on its preceding term(s).
In the provided exercise, we observe recursion with the formula:

• \( a_{k+1} = \frac{1}{2}a_k \)

For each number in the sequence, you take the term before it, multiply it by \(\frac{1}{2}\), and get your new term. This recursive rule paints a picture of how previous values determine subsequent ones, a hallmark of recursive processes. The main idea is simplicity through repetition of a formula that references itself.

Precalculus Concepts

In precalculus, understanding sequences and recursion lays the groundwork for tackling more advanced topics in calculus and algebra. Sequences typically represent series or uniform increments seen frequently in calculus problems. Recursion becomes a valuable tool for investigating patterns and functions more intuitively and iteratively.
Precalculus students are often required to identify initial terms, apply recursive formulas, and generate terms in sequences. This not only illustrates how mathematical formulas operate but also sharpens logical thinking and problem-solving skills. The exercise highlights these precalculus concepts by emphasizing:

• Initial term recognition, establishing starting value for recursive calculations.
• Application of recursive formulas to extend a sequence.
• Understanding diminishing sequences, demonstrated in the half-life of the terms.

Such exercises prepare students for the complexity of calculus by reinforcing both computational skills and conceptual understanding.