Problem 108 - Algebra and Trigonometry

Question

In Exercises \(105-108\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\)

Step-by-Step Solution

Answer

Yes, the statement is true. The common ratio of the geometric progression for the given \(a_n = 3(0.5)^{n-1}\) is indeed \(\frac{1}{2}\).

Step 1: Understanding the problem

The statement is about a geometric progression and we are given the nth term of the progression as \(a_n = 3(0.5)^{n-1}\). Our task is to examine if the common ratio of this progression is \(\frac{1}{2}\).

Step 2: Identification of variables in the formula

In the standard formula for a geometric sequence \(a_n = a_1 \times r^{n-1}\), the first term \(a_1\) can be obtained by substituting n as 1 in the given nth term formula. So, \(a_1 = 3(0.5)^{1-1} = 3\).

Step 3: Comparison with the standard formula

Now, if we compare our given formula \(a_n = 3(0.5)^{n-1}\) with the standard form, we can see that \(a_1\) is 3 and \(r\) is \(0.5 = \frac{1}{2}\). So, it matches with our calculated values and the statement is true.

Key Concepts

Common Ratio

The common ratio in a geometric sequence is a key component that defines the progression of the sequence. This ratio, often denoted by the letter 'r', is the factor by which any term of the sequence is multiplied to get the subsequent term. For example, in the sequence 2, 4, 8, 16, the common ratio is 2 because each term is twice the previous one.

In the exercise provided, the geometric sequence is expressed as an nth term formula, which is a common way to represent a sequence. The formula provided is \( a_n = 3(0.5)^{n-1} \). From this, we can deduce that the common ratio 'r' is 0.5, which also equates to \( \frac{1}{2} \) when expressed as a fraction. This is the constant factor that each term of the sequence is multiplied by to reach the next term, confirming that for any geometric sequence, understanding the common ratio is essential.

Moreover, it's crucial to recognize that the common ratio should not be zero because this would make every term after the first one zero, resulting in a rather uninteresting sequence.

Nth Term

The concept of the nth term is central to understanding any sequence, as it provides a way to directly calculate any term in the sequence based on its position. This particular component is especially useful when dealing with long sequences, where manually counting each term can become impractical.

In a geometric sequence, the nth term formula has the general form \( a_n = a_1 r^{n-1} \), where \( a_1 \) is the first term of the sequence and 'r' is the common ratio. To find the nth term, you simply plug the values of \( a_1 \) and 'r' into the formula along with the specific position 'n' you're interested in.

Referring to our example, the nth term is given by \( a_n = 3(0.5)^{n-1} \). If you wanted to find the 5th term, for instance, you would calculate it as \( a_5 = 3(0.5)^{5-1} = 3(0.5)^4 = 3(\frac{1}{16}) = \frac{3}{16} \). It's this formula that allows quick access to any term within a geometric sequence without the need to list out all the preceding terms.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence shows exponential growth or decay depending on whether the common ratio is greater than or less than one, respectively.

For example, the sequence \( 2, 6, 18, 54, ... \) is a geometric progression with a common ratio of 3. Similarly, the sequence in the exercise, given by the nth term formula \( a_n = 3(0.5)^{n-1} \), represents a geometric progression where each term decreases by half the previous term, indicative of a common ratio less than one.

Geometric progressions have vast applications across various fields including physics, biology, economics, and finance, particularly in modeling situations where growth or decay happens at a consistent relative rate. Understanding the nature of geometric sequences allows for better comprehension of phenomena that follow this pattern, from the decay of radioactive substances to the growth of investments.