Problem 39 - 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. $$ -16,-8,-4,-2, \ldots $$

Step-by-Step Solution

Answer

The next two terms in the sequence are -1 and -0.5. The explicit formula is \(a_n = -32 / (2^n)\) and the recursive formula is \(a_n = a_{n-1}/2\) for \(n > 1\) with the first term \(a_1 = -16\).

Step 1: Identify the pattern of the sequence

First, recognise the pattern or rule of the sequence. Looking at the given sequence \( -16, -8, -4, -2 \), it can be concluded that each term is half of the previous term.

Step 2: Find the next two terms

Following the identified pattern, the next term is half of the last term of the sequence. Therefore, the next term after -2 will be \(-2 / 2 = -1\). Continuing the pattern, the term after -1 is \(-1 / 2 = -0.5\). So, the next two terms of the sequence are -1 and -0.5.

Step 3: Formulate explicit and recursive formulas

For the explicit formula, you might notice that the \(n^{th}\) term of the sequence is \(a_n = -32 / (2^n)\). This formula determines the \(n^{th}\) term regardless of the previous ones. On the other hand, the recursive formula can be expressed as \(a_n = a_{n-1}/2\) for \(n > 1\) and \(a_1 = -16\). This formula determines the \(n^{th}\) term based on the previous one.

Key Concepts

Explicit Formula

In the study of sequences, an explicit formula is a powerful tool. It allows us to find any term in the sequence without needing to reference the previous terms.
For the sequence given in the exercise, we identified an explicit formula as:\[ a_n = \frac{-32}{2^n} \]This formula tells us directly what the value of any term in the sequence is, just by knowing its position, denoted by\( n \).

• -32: This is the starting point, or the first adjustment factor in our formula. It reflects the sequence's initial term's relationship to its position.
• 2^n: The denominator reflects the geometric progression of the sequence. It shows the pattern that each term is obtained by an operation (in this case, division by 2) relative to its position.

This type of formula is great because it provides a clear and efficient way to determine any term in the sequence without computing all earlier terms.

Recursive Formula

A recursive formula, unlike an explicit formula, defines each term by referring to previous terms in the sequence. It is like building each term step by step, starting from the first.
For the sequence at hand, the recursive formula is defined as:\[ a_n = \frac{a_{n-1}}{2} \quad \text{for} \ n > 1 \quad \text{and} \quad a_1 = -16 \]This formula indicates two important things:

• Initial Term: The sequence begins with \(a_1 = -16\), which serves as the foundation for all subsequent calculations.
• Recursive Development: Each subsequent term is obtained by halving the previous term, showing a consistent operation applied to produce the sequence.

While recursive formulas may sometimes require you to compute all previous terms to find the desired term, they are particularly useful for understanding the sequence's construction and verifying its progression.

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. This difference is often referred to as the common difference.
In contrast, the sequence we're analyzing in the exercise is not arithmetic, because the difference is not consistent.Instead, each term is half the previous term, which makes it a geometric progression rather than an arithmetic one.

• Arithmetic Example: A true arithmetic sequence could look like \(-16, -12, -8, -4 \ldots\), where the common difference is 4.
• Not Applicable: Since our sequence computes each term by multiplying the previous term by a constant (0.5 for division by 2), it does not fit the arithmetic pattern.

It's important to differentiate these sequence types, as both arithmetic and geometric progressions can appear similar at a glance but have fundamentally different properties and formulas.