Problem 64 - 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.$$\sum_{n=0}^{6} 500(1.04)^{n}$$

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is approximately \$3494.92

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

The first term (a) of the sequence is 500. The common ratio (r) is 1.04 and the number of terms (n) is 7 (remember, it starts from 0).

Step 2: Use the formula

The sum \( S_n \) of the first n terms of a geometric sequence is given by the formula \( S_n = a \left( \frac{1 - r^n}{1 - r} \right) \), if the ratio is not 1.

Step 3: Substitution of given values

Substitute the given values into the formula. So, \( S_7 = 500 \left( \frac{1 - (1.04)^7}{1 - 1.04} \right) \).

Step 4: Calculate

Calculate to find the answer. Here, please note to apply the BODMAS rule which states that operations should be carried out in the following order: Brackets, Orders, Division and Multiplication (from left to right), Addition and Substraction (from left to right).

Step 5: Verification

Use a graphing utility to graph the sequence and verify that the sum of the terms is indeed correct

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. For example, in the sequence 2, 4, 8, 16, ..., the common ratio is 2, as each term is twice the previous one. Distinguishing characteristics of geometric sequences are exponential growth or decay, which is heavily determined by the common ratio. When the common ratio is greater than 1, the sequence grows, and when it is between 0 and 1, the sequence decays.

Geometric Series Formula

Geometric series involve the summing of the terms in a geometric sequence. The formula for the sum of the first n terms of a geometric series, denoted as \( S_n \), is given by \[ S_n = a \left( \frac{1 - r^n}{1 - r} \right) \], where 'a' is the first term of the sequence, 'r' is the common ratio, and 'n' is the number of terms to be summed, provided that the common ratio \( r eq 1 \). This formula helps in finding the sum quickly without needing to manually add all the terms.

Sum of a Finite Geometric Series

To calculate the sum of a finite geometric series, one applies the geometric series formula. For the exercise \( \sum_{n=0}^{6} 500(1.04)^{n} \), the sum \( S_7 \) is calculated by identifying the first term 'a' as 500, the common ratio 'r' as 1.04, and the number of terms 'n' as 7. The substitution yields \( S_7 = 500 \left( \frac{1 - (1.04)^7}{1 - 1.04} \right) \). Following the order of operations, the exponential part is calculated first, followed by the subtraction and division, finally leading to the sum of this specific geometric series.

Graphing Utility Verification

After calculating the sum algebraically, it's beneficial to verify the answer using a graphing utility. Graphing utilities plot the terms of the series on a coordinate plane, allowing visualization of the sequence's growth. By summing the plotted points, one can confirm the accuracy of the calculated sum. This step is important not only for verification but also for understanding the graphical representation of geometric sequences. It reinforces the concept that the sequence of partial sums of a geometric series will form a curve approaching a finite limit, provided the common ratio is between 0 and 1 (exclusive).