Problem 56 - 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. $$\sum_{n=1}^{9}(-2)^{n-1}$$

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is 341

Step 1: Understand the sum of a geometric progression

The formula for the sum of the first \(n\) terms \(S_n\) of a geometric progression is \(S_n = a(1 - r^n) / (1 - r)\) where \(a\) is the first term and \(r\) is the common ratio.

Step 2: Identify the common ratio and first term

From \((-2)^{n-1}\), it can be seen that \(r = -2\) is the common ratio and the first term \(a\) when \(n = 1\) is \((-2)^{0} = 1\).

Step 3: Apply the formula

Substitute the known values into the formula: \(S_9 = 1(1 - (-2)^9) / (1 - (-2)) = 341\).

Step 4: Verification with graphing utility

Use a graphing utility to plot the sum of the geometric sequence up to the 9th term. The result should corroborate the calculated sum.

Key Concepts

finite geometric series

A finite geometric series is the sum of a certain number of terms in a geometric sequence. A geometric sequence, also known as a geometric progression, involves multiplying by a constant factor, known as the common ratio, to get from one term to the next. When we talk about a finite geometric series, we're interested in summing these terms up to a certain point.

With finite geometric series, the number of terms you're summing is limited or finite. For example, in our exercise, we are looking at the series from the first term to the ninth term. Each term in the series follows the geometric pattern established by the common ratio. By understanding this pattern and using specific mathematical tools, we can find neatly the total of all these terms.

The finite series formula is really powerful. It allows you to find the sum easily without having to add up each term manually. This becomes especially handy with longer or more complex sequences.

sum of geometric sequence

The sum of a geometric sequence up to a specific number of terms is sometimes referred to as a finite geometric sum. This can be calculated efficiently with a well-known formula. For a geometric sequence, you have to know two key pieces of information:

• The first term of the sequence, often denoted as \(a\).
• The common ratio \(r\), which each term is multiplied by to get to the next.

This formula for the sum of the first \(n\) terms is given by:\[S_n = a \frac{1 - r^n}{1 - r}\] This neat formula helps you find the sum without adding each term separately. In context of our exercise, you substitute \(a\) and \(r\) into the formula to find the sum. This technique is both time-saving and allows you to easily verify results with graphical tools or manual calculations.

Understanding this formula is crucial for mastering sequences and series in mathematics. It breaks down the seemingly complex task of addition into a straightforward calculation.

geometric progression

A geometric progression, or 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 called the common ratio. For instance, in the series involving \((-2)^{n-1}\), the common ratio is \(-2\). This means each term is the previous term multiplied by \(-2\).

Let's break down some key terms:

• First term \(a\): It's the initial value of the sequence, defined here as 1 when \(n = 1\).
• Common ratio \(r\): It determines how the sequence progresses from one term to the next. A positive or negative ratio changes the behavior and appearance of the progression.

If the common ratio's absolute value is more than 1, the terms in the sequence increase (or decrease negatively) quickly. If the ratio is between \(-1\) and \(1\) exclusive, the terms gradually approach zero. Understanding these properties helps in exploring the behavior of sequences in various contexts.