Problem 10 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{i=1}^{6}(2 i-3)$$

Step-by-Step Solution

Answer

The sum is 24.

Step 1 - Understand the Summation Notation

The given series is written in summation notation: \( \sum_{i=1}^{6}(2i-3) \). This means we need to consider the expression \( 2i-3 \) for each value of \( i \) from 1 to 6, then sum these values.

Step 2 - Generate the Terms

Substitute each value of \( i \) from 1 to 6 into the expression \( 2i-3 \): For \( i = 1 \): \( 2(1) - 3 = -1 \) For \( i = 2 \): \( 2(2) - 3 = 1 \) For \( i = 3 \): \( 2(3) - 3 = 3 \) For \( i = 4 \): \( 2(4) - 3 = 5 \) For \( i = 5 \): \( 2(5) - 3 = 7 \) For \( i = 6 \): \( 2(6) - 3 = 9 \)

Step 3 - Summing the Terms

Now sum the generated terms: \( -1 + 1 + 3 + 5 + 7 + 9 \)

Step 4 - Calculate the Result

Perform the addition: \( -1 + 1 = 0 \) \( 0 + 3 = 3 \) \( 3 + 5 = 8 \) \( 8 + 7 = 15 \) \( 15 + 9 = 24 \)

Key Concepts

series summation

Summation notation is a way to compactly represent the sum of a sequence of numbers. The given example uses the summation symbol, \( \sum \), to indicate that we need to add up a series of values.
In the exercise, the notation \( \sum_{i=1}^{6}(2i-3) \) means that we take the expression \( 2i-3 \), evaluate it for each integer value of \( i \) from 1 to 6, and then sum all of these results together.
Summation notation is quite powerful and helps simplify lengthy arithmetic expressions. By understanding how to interpret and generate the terms in the summation, you can handle a wide variety of series problems.

arithmetic series

An arithmetic series is a series of numbers in which the difference between any two successive terms is constant. In the example, after generating the terms, we have: \[ -1, 1, 3, 5, 7, 9 \].
Here, the difference between each consecutive term is 2, making it an arithmetic series.

To find the sum of an arithmetic series, one method is to sum up each term individually, as shown in the exercise. Another method involves using the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.
For our series: \[ a_1 = -1, \, a_n = 9, \, n = 6 \]
So, the sum is: \[ S_6 = \frac{6}{2}(-1 + 9) = 3 \times 8 = 24 \], matching the calculated result from our exercise.

algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. The expression in our series \( 2i - 3 \) is a linear algebraic expression in terms of \( i \).
To solve problems involving such expressions, you can substitute different values for the variable and then perform the operations indicated. For example:

• For \( i = 1 \): \( 2(1) - 3 = -1 \)
• For \( i = 2 \): \( 2(2) - 3 = 1 \)
• For \( i = 3 \): \( 2(3) - 3 = 3 \)
• For \( i = 4 \): \( 2(4) - 3 = 5 \)
• For \( i = 5 \): \( 2(5) - 3 = 7 \)
• For \( i = 6 \): \( 2(6) - 3 = 9 \)

This results in the series \( -1, 1, 3, 5, 7, 9 \), which you then add up. Understanding how to manipulate and evaluate algebraic expressions is fundamental in algebra and crucial for solving more complex mathematical problems.