Problem 16 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=1, r=0.5 $$

Step-by-Step Solution

Answer

The explicit formula for the given sequence is \(a_{n} = 0.5^{n-1}\). The first five terms of the sequence are 1, 0.5, 0.25, 0.125 and 0.0625.

Step 1: Write Down the Geometric Sequence Formula

The general formula for a geometric sequence is \(a_{n} = a_{1} \cdot r^{(n-1)}\), where \(a_{n}\) is the nth term, \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Substitute Given Values in the Formula

Substitute \(a_{1} = 1\) and \(r = 0.5\) into the formula. Therefore, the explicit formula for this sequence is \(a_{n} = 1 \cdot (0.5)^{n-1} = 0.5^{n-1}\).

Step 3: Generate the First Five Terms

Using the explicit formula \(a_{n} = 0.5^{n-1}\), generate the first five terms of the sequence: \n1. For n=1, \(a_{1} = 0.5^{1-1} = 0.5^0 = 1\) \n2. For n=2, \(a_{2} = 0.5^{2-1} = 0.5^1 = 0.5\) \n3. For n=3, \(a_{3} = 0.5^{3-1} = 0.5^2 = 0.25\) \n4. For n=4, \(a_{4} = 0.5^{4-1} = 0.5^3 = 0.125\) \n5. For n=5, \(a_{5} = 0.5^{5-1} = 0.5^4 = 0.0625\)

Key Concepts

Explicit Formula

An explicit formula is a concise way to describe the entire sequence. It allows you to calculate any term in the sequence without needing to determine the preceding ones. This makes it very efficient, especially for large sequences. For our geometric sequence, the explicit formula is given by \(a_{n} = a_{1} \cdot r^{(n-1)}\). Here, \(a_{1}\) is the first term, and \(r\) is the common ratio, which we'll discuss shortly.
In our example, the explicit formula becomes \(a_{n} = 0.5^{n-1}\). This transformation happens by substituting the first term, 1, and the common ratio, 0.5. This formula gives us a direct way to find any term number \(n\) in the sequence.
When using an explicit formula, always double-check your calculations. It's quick and effective for calculation, but mistakes can lead to incorrect terms.

Common Ratio

The common ratio is a crucial part of any geometric sequence. It's the factor that we multiply or divide by to move from one term to the next. All terms in the geometric sequence are related by this constant factor. You find the common ratio \(r\) by dividing any term by the preceding term.
In our sequence, the common ratio is 0.5. This means each term is half of the previous term. It's this value that helps define the pattern and direction of the sequence. A common ratio less than 1, like ours (0.5), indicates a decreasing sequence.

• Positive ratios keep the sequence sign unchanged.
• Negative ratios cause the sequence to alternate between positive and negative terms.

Understanding this concept can make analyzing a sequence much easier.

Sequence Terms

Sequence terms are the individual elements within a sequence. In a geometric sequence like ours, they're generated using the explicit formula. This sequence begins with the first term, often referred to as \(a_{1}\). Subsequent terms are generated by consistently applying the common ratio.
For example, the first five terms of our sequence are: 1, 0.5, 0.25, 0.125, and 0.0625. We derive these values by substituting successive values of \(n\) into the explicit formula \(a_{n} = 0.5^{n-1}\).
Each sequence term helps in visualizing how the sequence behaves over time. By examining sequence terms, we can also understand trends or limits, like whether a sequence converges towards something. Recognizing these patterns supports deeper comprehension of how geometric sequences function.