Problem 64 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5}(8 i-1)$$

Step-by-Step Solution

Answer

The sum of the series is 115.

Step 1: Understand the Series

The expression for the series is given as \( \sum_{i=1}^{5}(8i-1) \). This means we need to evaluate the expression \( 8i - 1 \) for each integer \( i \) from 1 to 5, and then sum all these values.

Step 2: Compute Each Term

Substitute each value of \( i \) from 1 to 5 into the expression \( 8i - 1 \):- For \( i = 1 \): \( 8(1) - 1 = 7 \)- For \( i = 2 \): \( 8(2) - 1 = 15 \)- For \( i = 3 \): \( 8(3) - 1 = 23 \)- For \( i = 4 \): \( 8(4) - 1 = 31 \)- For \( i = 5 \): \( 8(5) - 1 = 39 \)

Step 3: Compute the Sum

Add the computed terms together to find the total sum: \[ 7 + 15 + 23 + 31 + 39 \].Calculate step by step:- \( 7 + 15 = 22 \)- \( 22 + 23 = 45 \)- \( 45 + 31 = 76 \)- \( 76 + 39 = 115 \)

Step 4: Write the Final Answer

The sum of the series is 115.

Key Concepts

Series Evaluation

In mathematics, evaluating a series is the process of calculating the sum of a sequence of numbers that are expressed in a specific pattern. The notation \( \sum \) represents the summation, guiding us through the calculation. In our exercise, we have the series \( \sum_{i=1}^{5}(8i-1) \), where \( i \) ranges from 1 to 5.
The series is practically a list of numbers generated by substituting each \( i \) value from the set \{1, 2, 3, 4, 5\} into the expression \( 8i - 1 \). For each \( i \), a term is calculated, resulting in numbers that we later sum up.
To evaluate any given series:

• Understand the expression pattern by identifying the formula involved.
• Substitute all appropriate values as instructed by the series notation.
• Compute each value precisely, and finally add them together to yield the sum.

This systematic approach ensures clarity and accuracy in evaluating series.

Arithmetic Series

An arithmetic series is a sum of terms that follow an arithmetic sequence. An arithmetic sequence is a sequence of numbers where each term after the first is derived by adding a constant difference. This is an important concept to grasp, as it connects to various mathematical aspects.
In the exercise, the expression \( 8i - 1 \) results in the terms 7, 15, 23, 31, and 39. Notice how each term is increased by a constant value (8 in this case, derived from the coefficient of \( i \)). This assessment is what identifies the sequence as arithmetic.

• The starting term \( a \), here 7, sets up the sequence.
• The common difference, \( d \), is found by subtracting consecutive terms (\( 15 - 7 = 8 \) confirms this).
• The sum of an arithmetic series can also be calculated as \( S_n = \frac{n}{2} (a + l) \), where \( n \) is the total number of terms, \( a \) is the first term, and \( l \) is the last term. For our series, \( n = 5 \), \( a = 7 \), \( l = 39 \), providing another method to confirm our solution as 115.

Understanding these properties deepens comprehension and broadens problem-solving strategies.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and arithmetic operations. In the exercise, the expression \( 8i - 1 \) illustrates this concept by involving:

• Numerical coefficients (in this case, 8, which is a multiplier for the variable \( i \)).
• Variables, typically represented by letters like \( i \), which signify numbers that can change based on the context.
• Operations, such as subtraction here, which organize how the components interact to form results.

Algebraic expressions are tools for both generalization and specific calculations. They allow us to create generalized formulas applicable to broad scenarios and enable us to plug specific values for detailed solutions.
Practicing with algebraic expressions helps develop the skill to transform words or real-world scenarios into mathematical models, effectively placing problem-solving in a systematic framework.