Problem 6
Question
In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{45} 3$$
Step-by-Step Solution
Verified Answer
The evaluated sum is 135.
1Step 1: Identify the Summation
Recognize that we are summing a constant, 3, from i = 1 to 45. This simply means we are adding up the number 3, 45 times.
2Step 2: Apply the Summation Formula
The formula \(\sum_{i=1}^{n} c = nc\) is used when you are summing a constant 'c', 'n' times. Apply this formula by substituting 'n' with 45 and 'c' with 3 to get \(45 * 3\).
3Step 3: Calculate the Sum
Multiply 45 and 3 to get the sum, 135.
Key Concepts
Constant Series SumSigma NotationArithmetic Series
Constant Series Sum
When faced with the task of adding up a long list of the same number, the concept of a constant series sum comes into play. This is a straightforward notion in mathematics where you have a sequence of identical numbers, known commonly as terms, and you need to find the total sum of these terms. For example, if you want to add the number 3 together 45 times, instead of writing out 3 + 3 + 3 + ... (45 times), we can simplify the process using the constant series formula.
The general formula for the sum of a constant series is given by:
\[ S = n \times c \] where \( S \) is the sum of the series, \( n \) is the number of terms, and \( c \) is the constant number you're adding. In the exercise, we use this formula by setting \( n = 45 \) (the number of terms) and \( c = 3 \) (the constant number), resulting in \( S = 45 \times 3 \), which equals 135. This is how you effortlessly find the sum of a constant series, saving both time and effort, especially with larger numbers of terms.
The general formula for the sum of a constant series is given by:
\[ S = n \times c \] where \( S \) is the sum of the series, \( n \) is the number of terms, and \( c \) is the constant number you're adding. In the exercise, we use this formula by setting \( n = 45 \) (the number of terms) and \( c = 3 \) (the constant number), resulting in \( S = 45 \times 3 \), which equals 135. This is how you effortlessly find the sum of a constant series, saving both time and effort, especially with larger numbers of terms.
Sigma Notation
The symbol used to denote the summation of a series, \( \Sigma \), is referred to as sigma notation. It's a concise and powerful mathematical convention for representing sums. Sigma, the Greek letter \( \Sigma \), is used to indicate that you should 'sum up' whatever expression follows it, over the range of values specified.
In sigma notation, a series is written in the form: \[ \sum_{i=m}^{n} a_i \] where \( i \) is the index of summation, starting at the lower limit \( m \) and ending at the upper limit \( n \). The term \( a_i \) represents the sequence of numbers to be added. The expression underneath the sigma symbol states where the summation starts, and the one above tells us where it ends.
For the current exercise, the sigma notation is \( \sum_{i=1}^{45} 3 \), indicating that we add the constant number 3 for each value of \( i \) from 1 through 45. Understanding sigma notation allows for a uniform way to express complex sums and simplifies the process of communicating mathematical concepts.
In sigma notation, a series is written in the form: \[ \sum_{i=m}^{n} a_i \] where \( i \) is the index of summation, starting at the lower limit \( m \) and ending at the upper limit \( n \). The term \( a_i \) represents the sequence of numbers to be added. The expression underneath the sigma symbol states where the summation starts, and the one above tells us where it ends.
For the current exercise, the sigma notation is \( \sum_{i=1}^{45} 3 \), indicating that we add the constant number 3 for each value of \( i \) from 1 through 45. Understanding sigma notation allows for a uniform way to express complex sums and simplifies the process of communicating mathematical concepts.
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which the difference between consecutive terms is constant. This concept is slightly different from the constant series sum as the arithmetic series involves adding terms that have a common difference between them.
The sum of an arithmetic series can be calculated quickly using the formula: \[ S = \frac{n}{2} (a_1 + a_n) \] where \( S \) is the sum of the series, \( n \) is the total number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. A faster method for series with a large number of terms involves using the average of the first and last term, multiplied by the number of terms.
Understanding the difference between a constant series sum and an arithmetic series is crucial for correctly solving various kinds of summation problems. While in the exercise provided we dealt with a constant series sum, being familiar with arithmetic series is essential for a well-rounded grasp of summation topics in mathematics.
The sum of an arithmetic series can be calculated quickly using the formula: \[ S = \frac{n}{2} (a_1 + a_n) \] where \( S \) is the sum of the series, \( n \) is the total number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. A faster method for series with a large number of terms involves using the average of the first and last term, multiplied by the number of terms.
Understanding the difference between a constant series sum and an arithmetic series is crucial for correctly solving various kinds of summation problems. While in the exercise provided we dealt with a constant series sum, being familiar with arithmetic series is essential for a well-rounded grasp of summation topics in mathematics.
Other exercises in this chapter
Problem 5
In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{60} 7$$
View solution Problem 5
GEOMETRY You create an open box from a square piece of material 24 centimeters on a side. You cut equal squares from the corners and turn up the sides. (a) Draw
View solution Problem 6
GEOMETRY You are given wire and are asked to forma right triangle with a hypotenuse of \(\sqrt{18}\) inches whose area is as large as possible. (a) Draw and lab
View solution Problem 7
In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{20} i^3$$
View solution