Problem 40 - Beginning and Intermediate Algebra

Question

Evaluate each series. \sum_{i=1}^{5}(4 i+3)

Step-by-Step Solution

Answer

The sum of the series \(\sum_{i=1}^{5}(4i+3)\) is equal to 75.

Step 1: Write out the explicit form of the first 5 terms.

First, we want to write out the first 5 terms of this series by substituting the values of \(i\) from 1 to 5 into the general form of each term: 1st term: \(4(1)+3=7\) 2nd term: \(4(2)+3=11\) 3rd term: \(4(3)+3=15\) 4th term: \(4(4)+3=19\) 5th term: \(4(5)+3=23\) The first 5 terms of the series are: 7, 11, 15, 19, 23.

Step 2: Calculate the sum of the terms.

Now, we can calculate the sum of these terms. Sum = 1st term + 2nd term + 3rd term + 4th term + 5th term Sum = 7 + 11 + 15 + 19 + 23

Step 3: Perform the calculations.

Now we perform the addition: Sum = 7 + 11 + 15 + 19 + 23 = 75 So, the sum of the series \(\sum_{i=1}^{5}(4i+3)\) is equal to 75.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two successive terms is constant. This difference is called the common difference. In the provided exercise, each term is formed by adding a consistent amount to the previous term. For example, the terms generated from the expression \(4i+3\) form an arithmetic sequence. When \(i\) is increased by 1, each subsequent term increases by 4. Hence, the common difference is 4.

• 1st term: \(4(1)+3 = 7\)
• 2nd term: \(4(2)+3 = 11\)
• 3rd term: \(4(3)+3 = 15\)
• 4th term: \(4(4)+3 = 19\)
• 5th term: \(4(5)+3 = 23\)

Understanding the nature of an arithmetic series is essential in evaluating its sum efficiently.

Sum of Series

The sum of a series involves adding all the terms in the sequence. For arithmetic series like the one in the exercise, there is a handy formula to calculate the sum, but for small series, it's often simpler to add the numbers directly as demonstrated. The sum of the given series is determined by totaling the values of the terms:

• 7 + 11 + 15 + 19 + 23

When you add these numbers, the sum reaches 75. This calculation can be verified using the arithmetic series sum formula:
\[S_n = \frac{n}{2} \times (a + l)\]Where \(S_n\) is the sum of the series, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.
Here, \(n = 5\), \(a = 7\), and \(l = 23\). Thus:\(S_5 = \frac{5}{2} \times (7+23) = 75\). This confirms the manual addition was correct.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operational symbols. In solving series problems, understanding these expressions is crucial. For instance, the expression \(4i+3\) in the given series is an algebraic representation where "\(i\)" is the variable component that changes its value in integer steps from 1 to 5. The operations involve multiplication and addition.
These expressions are the basis for creating the sequence terms:

• For \(i=1\), the term is \(4(1)+3 = 7\)
• As \(i\) varies, each term can be evaluated similarly using the same expression.

Algebraic expressions compactly represent relationships and operations, making it easier to manage complex series.

Mathematical Calculations

Performing accurate mathematical calculations is key to evaluating the series. It involves substituting values into algebraic expressions and then executing the arithmetic operations of addition, multiplication, or division.
First, each term was calculated by following these steps:

• Substitute value of \(i\) into the algebraic expression \(4i+3\).
• Simplify the expression to obtain each term.

Finally, the total sum is calculated by adding all these terms. In this series, our sum becomes \(7 + 11 + 15 + 19 + 23 = 75\).
These mathematical calculations not only solve this series but also are widely applicable in various mathematical analyses, reinforcing the importance of understanding the underlying mechanical processes.