Problem 50 - Algebra 2

Question

What is the value of \(\sum_{n=1}^{5}(2 n-3) ?\) \(\begin{array}{lllll}{\text { F. } 6} & {\text { G. } 15} & {\text { H. } 17} & {\text { J. } 10 n-15}\end{array}\)

Step-by-Step Solution

Answer

The value of \(\sum_{n=1}^{5}(2 n-3)\) is 15.

Step 1: Identify the range and the summand

The range of the summation is all integer values from 1 to 5, inclusive. The series is given by \((2n - 3)\), so for each n in our range, we must evaluate \(2n - 3\).

Step 2: Calculate each term in the series

For each number \(n\) from 1 to 5, replace \(n\) in the summand with that number and calculate the result.Given \(2n - 3\):- for \(n = 1\), we get \(2(1)-3 = -1\).- for \(n = 2\), we get \(2(2)-3 = 1\).- for \(n = 3\), we get \(2(3)-3 = 3\).- for \(n = 4\), we get \(2(4)-3 = 5\). - for \(n = 5\), we get \(2(5)-3 = 7\).

Step 3: Sum all terms

Now sum all the calculated values to get the total series sum, which is \(-1 + 1 + 3 + 5 + 7 = 15\).

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers where each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is known as the "common difference."

When working with arithmetic series, understanding the general term is crucial. If you know the first term and the common difference, you can figure out any term in the series. By evaluating these, you can easily find the sum of the series.

In our exercise, the expression \(2n - 3\) describes the sequence. Even though the structure is not a classic constant sum pattern that arithmetic series usually depict, each evaluated result still forms a unique sequence when substituted for consecutive integer values of \(n\). The terms themselves in the series vary linearly, tracing a predictable pattern that results in a manageable summation process.

Algebraic Expressions

An algebraic expression is a mathematical statement composed of variables, numbers, and operations. In the exercise, the expression \(2n - 3\) is an algebraic expression.

Each part of the expression carries meaning:

• The coefficient \(2\) means that you multiply \(n\) by \(2\).


• The constant \(-3\) is subtracted after the multiplication, affecting every term in the series.

To evaluate an algebraic expression, substitute values for variables, then perform the indicated operations. Following the sequence, \(n\) values are substituted from \(1\) to \(5\), transforming the expression into understandable numbers. By simplifying each step, you ensure every calculation is correct, leading to the right summation.

Integer Sequences

The term "integer sequences" refers to ordered collections of whole numbers often seen in mathematical problems involving summation. Integer sequences can either be consecutive series like naturals or specified patterns defined by an algebraic rule.

In the exercise, integer sequences were formed by plugging successive numbers into the expression \(2n - 3\). The resulting sequence is \(-1, 1, 3, 5, 7\). These numbers show a straightforward pattern, each increasing by \(2\), aligning with arithmetic sequence behavior.

Recognizing and understanding integer sequences helps simplify complex mathematical operations by enabling predictions about future numbers. This ability transforms lengthy calculations into simple summations, much like the exercise where understanding the sequence was key to successfully summing the values.