Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{10}-k i $$

Step-by-Step Solution

Answer

The series sums to \(-55i\).

Step 1: Understanding the Series

The series given is \( \sum_{k=1}^{10} -k i \). This means we need to sum the terms from \( k = 1 \) to \( k = 10 \), where each term is \(-k i\).

Step 2: Write the Series as a Sum

We expand the series by substituting values of \( k \) from 1 to 10: \[ (-1)i + (-2)i + (-3)i + \, \ldots \, + (-10)i \]This can be written as \[ (-1)i + (-2)i + (-3)i + (-4)i + (-5)i + (-6)i + (-7)i + (-8)i + (-9)i + (-10)i \]

Step 3: Identify the Pattern and Simplify

Notice that each term in the series is \(-k i\). The real part is zero, and the imaginary part is the sum of negative integers from 1 to 10, which we can find by evaluating \(-1 - 2 - 3 - \ldots - 10\).

Step 4: Calculate the Arithmetic Sum

The sum of the integers from 1 to 10 is known as an arithmetic series, and can be calculated using the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2}(a + l) \] where \( n = 10 \), \( a = 1 \), and \( l = 10 \). So, \[ S_{10} = \frac{10}{2} (1 + 10) = 5 \times 11 = 55 \].

Step 5: Apply the Negative Sign

Since each term in the series is multiplied by \(-1\), we find the sum to be the negative of \( 55 \), i.e., \( -55 \). Thus, the sum of the series is \(-55i\).

Key Concepts

Sum of Arithmetic Series

An arithmetic series is a series of numbers in which each term increases by a constant amount, known as the common difference. The sum of an arithmetic series can be efficiently calculated using a simple formula, ensuring quick and accurate additions of terms.
To calculate the sum, you use the formula:

• \( S_n = \frac{n}{2}(a + l) \)
• Where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

In the example exercise, the arithmetic series sum was calculated using this formula. You need to identify the first term, the last term, and the number of terms. Then, apply the formula.
This approach avoids the need to add each term individually, especially useful for longer series.
In applications, it can be utilized in both everyday calculations and various fields such as finance for determining total payments or in physics for motion equations.

Complex Numbers

Complex numbers form a fundamental concept in mathematics that extends the idea of real numbers into a two-dimensional plane. They consist of a real part and an imaginary part. Expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
Complex numbers are useful in various fields, particularly in engineering and physics, to solve equations that traditional real numbers cannot.
A key feature of complex numbers is that they enable a more comprehensive understanding of equations that involve square roots of negative numbers. Their inclusion allows continuous and smooth solutions beyond the restrictions of real numbers. This makes them pivotal in calculations involving waves, oscillations, and systems analysis that occur in real-world scenarios.
Understanding and using complex numbers involves learning operations such as addition, subtraction, and multiplication on these numbers, as well as representing them graphically on the complex plane, enhancing both computation and interpretation.

Imaginary Numbers

Imaginary numbers are mathematical constructs that play a crucial role in advanced mathematical computations. They are defined as numbers that, when squared, yield a negative result. The most basic imaginary number is \( i \), where \( i^2 = -1 \).
Imaginary numbers aren't just theoretical; they have practical applications in engineering, physics, and technology, especially in areas that involve electricity and signal processing. Imaginary numbers, when combined with real numbers, form complex numbers, which help in solving polynomial equations that have no real roots.
They allow mathematicians and scientists to work with concepts where real number solutions would fall short, such as in understanding electromagnetic fields and fluid dynamics.
Learning about imaginary numbers often includes discovering how they fit into the broader category of complex numbers, their properties, and how they facilitate the resolution of real-world problems by extending the number system.