Problem 60 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. 2-4+8-16+32

Step-by-Step Solution

Answer

The summation notation for the given series is \(\sum_{n=1}^{5} 2 × (-2)^{n-1}\).

Step 1: Find the first term and the common ratio

The first term of the given series is 2. And since every term can be obtained by multiplying the previous term by -2, the common ratio is -2.

Step 2: General term formula

Using the common ratio, we can create a formula for the general term. The general term of a geometric series is given by \(a_n = a_1 × r^{n-1}\), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the number of terms. In this case, \(a_1 = 2\), and \(r = -2\). So, the general term formula is: \(a_n = 2 × (-2)^{n-1}\)

Step 3: Determine the range

As we only have information about the first 5 terms of the series, the range of our summation will be from n=1 to n=5.

Step 4: Write the summation notation

Using the general term formula and the determined range, we can write the given series using summation notation: \(\sum_{n=1}^{5} 2 × (-2)^{n-1}\)

Key Concepts

Geometric Series

A geometric series is a sequence of terms where each term after the first is found by multiplying the previous term by a constant, called the common ratio. This type of series can be identified by their pattern of consistent multiplication.
For example, in the series 2, -4, 8, -16, 32, each term is obtained by multiplying the previous term by -2. You start with the first term, 2, and continuously multiply it by the common ratio, -2, to find the subsequent terms. It's like a snowball effect, where each term builds off the last.

• First term: 2
• Second term: 2 × (-2) = -4
• Third term: -4 × (-2) = 8
• Fourth term: 8 × (-2) = -16
• Fifth term: -16 × (-2) = 32

This predictable pattern of growth or decrease allows us to represent the series neatly using summation notation.

General Term Formula

The general term formula is a powerful tool in expressing any term in a geometric series. It helps you calculate any term position in the series without having to list all previous terms. The formula for any term in a geometric series is given as:
\(a_n = a_1 \times r^{n-1}\)
Here:

• \(a_n\) is the nth term you're trying to find.
• \(a_1\) is the first term of the series.
• \(r\) stands for the common ratio, which is the factor you multiply by each time.
• \(n\) is the position of the term in the series.

So if you're trying to find any term in the series 2, -4, 8, -16, 32, you plug the known values into the formula.
For this series with \(a_1 = 2\) and \(r = -2\):
\(a_n = 2 \times (-2)^{n-1}\)
This formula enables you to calculate the nth term directly, which is particularly valuable with long series.

Common Ratio

The common ratio is a key component of a geometric series and is what links each term to the previous one. It is a constant value that each term is multiplied by to obtain the next term. Understanding the common ratio is crucial for identifying and working with geometric series.
To find the common ratio \(r\), take any term in the series and divide it by the previous term. In the series 2, -4, 8, -16, 32, you can calculate \(r\) as follows:

• Common ratio between -4 and 2: \(-4 \div 2 = -2\)
• Common ratio between 8 and -4: \(8 \div (-4) = -2\)
• Common ratio in subsequent terms: also -2

Consistent results of \(-2\) confirm the constant pattern, proving that the common ratio is indeed \(-2\). This multiplication factor is used in the general term formula, helping to describe and predict future terms in the series.