Problem 35 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i}$$

Step-by-Step Solution

Answer

The sum of the series \(\sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i}\) is \(-\frac{3}{2}\).

Step 1: Identify the type of series

The series follows the format \(a * r^{i}\), where \(a\) is the first element in the series, \(r\) is the common ratio between each consecutive terms and \(i\) identifies the position of the term. This confirm it's a geometric series.

Step 2: Identify the values of a and r

Looking at the series: \(\sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i}\), we observe that \(a\), which is the first term in the series is when \(i = 1\). Plugging that in gives us \(a= -\frac{1}{2}\). Also, as it is a geometric series, \(r\) is the common ratio between each terms, which is also \(-\frac{1}{2}\) in this case.

Step 3: Calculate the series

The standard formula to calculate the sum of a finite geometric series is \(\frac{a*(1-r^n)}{1-r}\) where a is the first term, r is the common ratio and n is the number of terms. Here, n equals to 2 (the number of terms in the series), a equals to \(-\frac{1}{2}\), and r also equals to \(-\frac{1}{2}\). When we insert these values into the formula, we compute \(\sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i} = \frac{-\frac{1}{2} * (1 - \left(-\frac{1}{2}\right)^2)}{1 - -\frac{1}{2}}\). After simplifying this, we get \(\frac{-3}{2}\)

Key Concepts

Finite Geometric Series

A finite geometric series is a sum of terms that share a common ratio, but unlike an infinite series, it stops after a certain number of terms. Each term in a geometric series is generated by multiplying the previous term by a fixed number. This fixed number is known as the common ratio, often denoted as \( r \). Finite geometric series can be represented as:

• \( a \) is the first term.
• \( n \) is the number of terms.
• \( r \) is the common ratio.

Understanding that a geometric series is finite is crucial because it allows us to determine the exact sum of all terms. The general formula for the sum of a finite geometric series is \( S_n = \frac{a(1-r^n)}{1-r} \). This formula sums the sequence quickly without having to add each individual term manually. The arithmetic in this formula handles the common ratio and number of terms simultaneously, giving us a precise total.

Sum of a Series

In mathematics, the sum of a series refers to the total when all terms in a sequence are added together. For geometric series, a specialized formula is used to find their sum efficiently. This formula for a finite geometric series, \( S_n = \frac{a(1-r^n)}{1-r} \), stands out because:

• Even with alternating signs or fractional terms, it gives the result directly.
• Makes it easier to perform complex sums that would otherwise be tedious.

Applying this formula requires careful identification of each component. We should correctly identify the first term \( a \), the common ratio \( r \), and the total number of terms \( n \). Each component defines the behavior of the sequence and influences the final outcome. By putting the values into the formula, it swiftly computes the sum.

Common Ratio

The common ratio is a pivotal element in a geometric series. It is the consistent factor that each term gets multiplied by to yield the next term in the sequence. This ratio can be a positive or negative number and determines the direction and nature of the series:

• If the ratio \( r \) is a fraction (between 0 and 1), each sequential term becomes smaller.
• If \( r \) is negative, the series will alternate in sign from term to term.

In the series \( \sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i} \), the common ratio is \(-\frac{1}{2}\). This negative ratio means each term alternates signs. Starting from the first term \( a \), which is \(-\frac{1}{2}\), the next term is achieved by multiplying by the common ratio again. Understanding the role of the common ratio helps in predicting the pattern of the series and applying the sum formula effectively.