Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=3}^{7}(5 i+2)$$

Step-by-Step Solution

Answer

The sum of the series is 135.

Step 1: Identify the Series Bounds

The series is represented as \(\sum_{i=3}^{7}(5i+2)\). This indicates that the variable \(i\) starts at 3 and ends at 7. The series will therefore include the terms calculated for \(i = 3, 4, 5, 6,\) and \(7\).

Step 2: Calculate Each Term in the Series

Substitute each value of \(i\) from the bounds into the expression \(5i + 2\).- For \(i = 3\): \(5(3) + 2 = 15 + 2 = 17\)- For \(i = 4\): \(5(4) + 2 = 20 + 2 = 22\)- For \(i = 5\): \(5(5) + 2 = 25 + 2 = 27\)- For \(i = 6\): \(5(6) + 2 = 30 + 2 = 32\)- For \(i = 7\): \(5(7) + 2 = 35 + 2 = 37\)

Step 3: Sum All Terms of the Series

Add all the calculated terms together to find the sum of the series:\(17 + 22 + 27 + 32 + 37 = 135\).

Key Concepts

Summation

Summation is a fundamental concept in algebra and arithmetic, often represented by the symbol \(\Sigma\). It denotes the process of adding a sequence of numbers. In this exercise, the aim is to sum the terms generated by the algebraic expression \(5i + 2\) where \(i\) ranges from 3 to 7. Summation helps us manage and simplify calculations involving numerous terms.

• The concept of summation is crucial in both pure and applied mathematics.
• It's used to find the total of a series of numbers.
• It aids in understanding patterns and results in arithmetic and geometric series.

Always ensure to accurately calculate each term and double-check your added results. This meticulous approach avoids common errors in problem-solving.

Algebraic Expression

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In our given exercise, the expression \(5i + 2\) is used to generate terms for our series. Algebraic expressions are invaluable tools for summarizing and analyzing mathematical data.

• An algebraic expression can model real-world situations, making them versatile in mathematics.
• They involve operations such as addition, subtraction, multiplication, and sometimes powers.
• When working with expressions, identifying terms and coefficients is vital to understanding their structure and effect.

Understanding how to manipulate these expressions seamlessly integrates into problem-solving in both arithmetic and algebraic contexts.

Finite Series

A finite series is the summation of a sequence that has a clear beginning and end. In this exercise, our series is finite because the variable \(i\) ranges from 3 to 7, giving us a limited number of terms to add. Understanding finite series is essential in many areas of mathematics and helps in analyzing data efficiently.

• Finite series contrast with infinite series, which continue indefinitely without a clear end point.
• They are easier to calculate because they involve a specific number of additions.
• Finite series are used in diverse fields, from statistics to computer science, to solve real-world problems.

Focusing on both the start and end of your series aids in ensuring all terms are correctly included and summed.