Problem 42 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=3}^{6}\left(2 i^{2}+1\right)$$

Step-by-Step Solution

Answer

The sum of the series is 176.

Step 1: Understand the Series

We are given a series \(\sum_{i=3}^{6}(2i^2 + 1)\). This notation means we need to find the sum of the expression \(2i^2 + 1\) from \(i = 3\) to \(i = 6\).

Step 2: Evaluate the Expression for each Term

Evaluate the expression \(2i^2 + 1\) for each integer value of \(i\) from 3 to 6. - For \(i=3\): \(2(3^2)+1 = 2\times9 + 1 = 19\).- For \(i=4\): \(2(4^2)+1 = 2\times16 + 1 = 33\).- For \(i=5\): \(2(5^2)+1 = 2\times25 + 1 = 51\).- For \(i=6\): \(2(6^2)+1 = 2\times36 + 1 = 73\).

Step 3: Calculate the Sum of the Evaluated Values

Add each of the evaluated values from Step 2 together: \(19 + 33 + 51 + 73\).

Step 4: Perform the Addition

Calculate the sum: - \(19 + 33 = 52\)- \(52 + 51 = 103\)- \(103 + 73 = 176\).Thus, the total sum of the series is 176.

Key Concepts

Understanding Summation

Summation is a fundamental concept in mathematics that involves adding up a sequence of numbers or expressions. Here, we deal with the summation indicated by the sigma notation, \(\sum\), which instructs us to sum the values of a given expression over a specified range of integer values. In our exercise, the series \(\sum_{i=3}^{6}\left(2 i^{2}+1\right)\) tells us to compute the expression \(2i^2 + 1\) for values of \(i\) starting from 3 up to 6.

• Start Value: The series starts at \(i = 3\).
• End Value: It ends at \(i = 6\).
• Expression: Each term to be summed is determined by \(2i^2 + 1\).

This process of summation ensures that the entire range is covered systematically. With summation, we get an aggregate value which is essential in both simple arithmetic and more complex mathematical analyses.

Exploring Algebra in Series

Algebra is a key tool for solving series problems, as it provides a structured way to evaluate mathematical expressions. At the heart of algebra in this context is the manipulation and evaluation of the expression \(2i^2 + 1\). For each value of \(i\) in the given range, algebra allows us to substitute and calculate results accurately. For instance, when \(i = 3\), the expression becomes \(2(3^2) + 1\) which simplifies to \(2\times 9 + 1\), giving 19. This algebraic substitution is performed for each value of \(i\):

• \(i = 4\): \(2(4^2) + 1 = 33\)
• \(i = 5\): \(2(5^2) + 1 = 51\)
• \(i = 6\): \(2(6^2) + 1 = 73\)

By understanding how to apply algebraic rules, students can focus on simplifying and solving expressions correctly across any arithmetic or polynomial series.

Expressions and Their Evaluation

Expressions in mathematics are combinations of symbols and numbers to denote computations. In solving series problems, such as the one presented, we frequently encounter algebraic expressions which must be evaluated step by step to solve the problem.The expression \(2i^2 + 1\) is an example of a polynomial expression involving squared terms. Here, each \(i\) is successively replaced with integers from 3 to 6. Evaluating these expressions involves straightforward arithmetic:

• Square \(i\): Calculate \(i^2\).
• Multiply by 2: Double the square of \(i\).
• Add 1: Finally, add 1 to the result.

By breaking down expressions into manageable parts, such evaluations become less daunting and more structured, allowing students to systematically derive the parts of the summation needed for complete calculation.