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Magazine Chapter 8: Problem 64
Find the sum for each series. $$\sum_{i=1}^{5}(8 i-1)$$
Short Answer
Expert verified
The sum of the series is 115.
Step by step solution
01
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.
02
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 \( i = 5 \): \( 8(5) - 1 = 39 \)
03
Compute the Sum
Add the computed terms together to find the total sum: \[ 7 + 15 + 23 + 31 + 39 \].Calculate step by step:- \( 7 + 15 = 22 \)- \( 22 + 23 = 45 \)- \( 45 + 31 = 76 \)- \( 76 + 39 = 115 \)
04
Write the Final Answer
The sum of the series is 115.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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.
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.
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:
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.
- 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.
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.
