Problem 54 - Algebra 2

Question

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=-2, a_{n}=\frac{1}{2} a_{n-1} $$

Step-by-Step Solution

Answer

The explicit formula for the given sequence is \(a_n=-2 \times (\frac{1}{2})^{n-1}\)

Step 1: Understanding the terms

Firstly, note that a recursive formula provides the next term of a sequence depending on the previous term. An explicit formula directly relates the term number with its specific value. The initial value \(a_1=-2\) will be of great help to devise the explicit formula.

Step 2: Analyzing the sequence

By using the recursive formula, generate a few more terms: \(a_1=-2\), \(a_2= \frac{1}{2}a_{1}\), \(a_3= \frac{1}{2}a_{2}\), \(a_4= \frac{1}{2}a_{3}\) etc. This helps to understand the pattern and trend of the sequence progression.

Step 3: Create the explicit formula

Notice that each term is half of the previous term and starts from the initial value. Hence, on each step, we keep dividing the initial number \(a_1\) by 2. The number of times we divide is \(n-1\) (since no division for \(n=1\) and we divide once for \(n=2\), and so on). The explicit formula derived is \(a_n=a_1 \times (\frac{1}{2})^{n-1}\)

Key Concepts

Explicit Formula

An explicit formula provides a way to find any term in a sequence without having to know the preceding terms. This is different from a recursive formula where you derive terms based on previous terms. Think of the explicit formula as giving you the whole picture at once. It's like having a direct map from your starting point to your destination, rather than a set of instructions that tell you step-by-step how to get there.

In our context, the explicit formula lets us figure out the value of a term based on its position in the sequence. For the given sequence, the explicit formula is found by identifying the pattern that relates the position of a term (noted as \( n \)) to its value. Here, the term \( a_n \) can be calculated using the formula:

• \( a_n = -2 \times (\frac{1}{2})^{n-1} \)

This formula suggests that knowing the term number \( n \), you can compute its value directly and efficiently without needing other terms.

Sequence Progression

Understanding sequence progression is key to recognizing patterns in sequences. When analyzing progression, we look at how terms develop from one to the next. This often involves observing whether the sequence grows, shrinks, or stays constant.

In the provided exercise, the sequence progression is guided by the rule where each term is half of the previous one. Starting from an initial term of \(-2\), each subsequent term decreases by a factor of \(\frac{1}{2}\). For example:

• \( a_1 = -2 \)
• \( a_2 = \frac{1}{2} \times a_1 = -1 \)
• \( a_3 = \frac{1}{2} \times a_2 = -0.5 \)
• \( a_4 = \frac{1}{2} \times a_3 = -0.25 \)

Here, you're seeing a geometric progression, which is why understanding that pattern helps in deriving the explicit formula.

Initial Value

The initial value is the starting point of the sequence. It's crucial in setting the path or trend for how the entire sequence will unfold. By knowing this initial value, you can predict subsequent terms using either a recursive formula or an explicit formula.

In our problem, the initial value \( a_1 \) is \(-2\). This initial value serves as the base from which all further terms are derived. Whether using the recursive rule or the explicit one, knowing \( a_1 \) ensures that you begin the sequence with the correct term. Often, the first term sets the initial condition required for a proper understanding of the formulae used later.

Recursive Formula

The recursive formula is like a blueprint for constructing a sequence one step at a time. It expresses each term based on its predecessor, thus requiring you to know past values to find future ones. This kind of formula is iterative, meaning you keep using the previous term to find the next one.

The recursive formula provided in this problem is:

• \( a_n = \frac{1}{2}a_{n-1} \)

This tells us that to find any term \( a_n \), you need to halve the term that came before it, \( a_{n-1} \). It's an incremental approach, building the sequence term by term.

This form might be cumbersome for finding far-off terms, which is why having an explicit formula acts as a shortcut to directly compute the values based on the term number alone.