Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of a Finite Geometric Sequence Find the sum. Use a graphing utility to verify your result. $$\sum_{n=0}^{5} 300(1.06)^{n}$$

Step-by-Step Solution

Answer

The sum of the given finite geometric sequence, which is calculated using the formula for the sum of a geometric sequence, can then be verified using a graphing utility.

Step 1: Identify the first term, common ratio, and number of terms

For the geometric sequence \(300(1.06)^n\), the first term (\(a\)) is 300, the common ratio (\(r\)) is 1.06, and the number of terms (\(n\)) is 6.

Step 2: Plug the identified values into the sum of geometric series formula

Apply the values you've found into the geometric sequence sum formula \(S_n = a(1 - r^n)/(1 - r)\). Therefore, \(S_6 = 300(1 - 1.06^6)/(1 - 1.06)\)

Step 3: Calculate the result

Doing the math for the formula above will give you the sum of the geometric sequence.

Step 4: Verify the result using a graphing utility

After finding the sum, verify the result using a graphing utility. Plot the function \(y = 300(1.06)^x\) for \(x = 0\) to \(x = 5\) and find the sum of y-values. This value should match the calculated sum.

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 4, 8, 16, ..., each term is obtained by multiplying the previous term by 2. The geometric sequence makes a pattern of points that can often be represented on a graph, showing exponential growth or decay.

To fully understand a geometric sequence, it's necessary to identify its first term, known as 'a', and the common ratio 'r'. From the exercise, the sequence is defined by the formula \(300(1.06)^n\), where 300 is the first term and 1.06 is the common ratio.

Common Ratio

The common ratio in a geometric sequence is pivotal as it dictates the progression of the sequence, whether it expands or contracts. This ratio is calculated by dividing any term by the preceding term, provided that there are no zero terms in the sequence. When the ratio is greater than 1, the sequence grows larger with each term; if it's between 0 and 1, the sequence decreases.

In our example, \(1.06\) is the common ratio. Since it is greater than 1, each subsequent term in the sequence will be 6% larger than the previous term, indicating a growing sequence. Understanding the common ratio is crucial before attempting to find the sum of the sequence.

Geometric Series Formula

The sum of the terms of a finite geometric sequence is known as a geometric series. To calculate this sum, there is a formula: \(S_n = \frac{a(1 - r^n)}{1 - r}\), where \(S_n\) is the sum of the first 'n' terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Using the provided exercise, where \(a = 300\), \(r = 1.06\), and \(n = 6\), we can plug these values into the formula to find the sum of the first six terms of the sequence. The formula helps to simplify the process of adding each term sequentially, saving time and reducing potential errors.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is an essential tool in mathematics, especially when working with sequences and series. It can plot the terms of a sequence as points on a graph, allowing for visual verification of properties like the common ratio and the behavior of a sequence.

When given a geometric sequence or series, like \(300(1.06)^n\), using a graphing utility can help check the sum obtained from the formula. By setting up the function and graphing for the given range of 'n', the consecutive y-values represent the terms of the sequence. Summing them gives another way to verify the result, which should align with the calculated sum. This method reinforces comprehension and confirms the sequence's behavior.