Problem 60 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{20} \frac{1}{2}$$

Step-by-Step Solution

Answer

The sum is 10.

Step 1: Identify the Proper Formula

The given series is a constant sum series. This means each term in the series is the same and equal to \( \frac{1}{2} \). The general formula to find the sum of a series where all terms are equal to a constant \( c \), and there are \( n \) terms, is \( S_n = n \cdot c \).

Step 2: Identify the Values of n and c

In this series, \( c = \frac{1}{2} \) because each term of the series is \( \frac{1}{2} \), and \( n = 20 \) because the series runs from 1 to 20. Therefore, we need to find the sum \( S_{20} \).

Step 3: Substitute the Values Into the Formula

Replace \( n \) with 20 and \( c \) with \( \frac{1}{2} \) in the formula: \[ S_{20} = 20 \times \frac{1}{2} \]

Step 4: Calculate the Sum

Compute the product: \[ S_{20} = 20 \times \frac{1}{2} = 10 \]Thus, the sum of the series is 10.

Key Concepts

constant series

A constant series is one where every term in the series is the same. Imagine you are adding up the same number repeatedly. This simplifies calculations quite a bit because you don't need to sum each term individually. Instead, you can use a straightforward approach to calculate the total.

For example, consider the series \( \sum_{i=1}^{20} \frac{1}{2} \). Each term here is \( \frac{1}{2} \), and there are 20 such terms. Since each term is the same, this is a classic example of a constant series.

In a constant series:

• Every term in the series has the same value.
• You can multiply the number of terms by the value of each term to find the sum.

This method saves a lot of time compared to adding each term one by one.

sum of series formula

To resolve a series efficiently, especially when dealing with a constant series, it’s essential to know the sum of series formula. This formula helps to quickly find the total value of a series when each term is identical.

The formula is \( S_n = n \cdot c \), where:

• \( n \) is the total number of terms.
• \( c \) is the constant value of each term.

For instance, in a constant series, such as our example \( \sum_{i=1}^{20} \frac{1}{2} \), we identify \( n = 20 \) and \( c = \frac{1}{2} \). By applying the formula, you multiply 20 by \( \frac{1}{2} \), resulting in 10.

This formula-backed method provides a quick solution without the need to painstakingly add each term, making it an invaluable tool for simplifying your work with series.

mathematical series

In mathematics, a series is a way of summing up a sequence of numbers. A series is simply the sum of the terms of a sequence. There are different types of series, such as arithmetic, geometric, and harmonic series, each with its own characteristics and formulas. However, the focus here is on the constant series.

A mathematical series can be defined by:

• The sequence of terms being summed.
• The formula used to find the sum.

For constant series, every term is the same, which means the series has a unique property where its sum can be easily calculated using multiplication.

Understanding the basics of mathematical series not only helps in solving school assignments but also lays the foundation for tackling more complex series problems later on. The principles learned here apply to various fields, from finance to physics, where series often play a critical role in analysis and computations.