Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 19
Use a formula to find the sum of the arithmetic series. $$ -7+(-4)+(-1)+2+5+\dots+98+101 $$
Short Answer
Expert verified
The sum of the series is 1739.
Step by step solution
01
Identify the First Term and Common Difference
The first term of the arithmetic series is clearly given as \(-7\). To find the common difference \(d\), we subtract the first term from the second term \((-4) - (-7) = 3\). Hence, \(a = -7\) and \(d = 3\).
02
Determine the Last Term and Use the Formula
The last term of the series is provided as \(101\). We will use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} \cdot (a + l) \]where \(l\) is the last term \(101\), and \(n\) is the number of terms.
03
Calculate the Number of Terms
Use the formula for the general term of an arithmetic sequence \(a_n = a + (n-1)d\) where \(a_n = 101\). Substitute the known values:\[ 101 = -7 + (n-1) \cdot 3 \]\[ 101 = -7 + 3n - 3 \]\[ 101 = 3n - 10 \]\[ 3n = 111 \]\[ n = \frac{111}{3} = 37 \]So, there are 37 terms in the series.
04
Substitute Values into the Sum Formula
Using the number of terms \(n = 37\), first term \(a = -7\), and last term \(l = 101\), substitute into the sum formula:\[ S_n = \frac{37}{2} \cdot (-7 + 101) \]\[ S_n = \frac{37}{2} \cdot 94 \]\[ S_n = 37 \times 47 \]\[ S_n = 1739 \]
05
Conclude the Sum of the Series
The calculated sum of the arithmetic series is \(1739\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Arithmetic Series
An arithmetic series is the sum of the terms in an arithmetic sequence. Calculating the sum involves knowing the pattern of the series so you can find a formula for quick calculation.
Each term in an arithmetic sequence is obtained by adding a constant difference to the previous term. To find the sum, we use the formula: \[ S_n = \frac{n}{2} \cdot (a + l) \] 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.
This formula comes from adding the sequence twice, forward and backward. This way we can easily see every number pairs with another to make a consistent sum.
Each term in an arithmetic sequence is obtained by adding a constant difference to the previous term. To find the sum, we use the formula: \[ S_n = \frac{n}{2} \cdot (a + l) \] 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.
This formula comes from adding the sequence twice, forward and backward. This way we can easily see every number pairs with another to make a consistent sum.
First Term
The first term of an arithmetic series is vital. It sets the starting point of the sequence.
In our example, the first term \(a\) is -7. This term is the initial number from which the series begins its progression.
The first term is crucial when applying any of the formulas related to arithmetic sequences or series.
In our example, the first term \(a\) is -7. This term is the initial number from which the series begins its progression.
The first term is crucial when applying any of the formulas related to arithmetic sequences or series.
- It helps to use the first term to easily calculate following terms, by adding the common difference repeatedly.
- In our series, all calculations regarding the sum or specific terms start with the first term.
Common Difference
The common difference in an arithmetic sequence is the amount that you add (or subtract) each time to get from one term to the next.
Understanding this value helps in predicting any term or summing up the series. In our example, the common difference \(d\) is 3. This means each term increases by 3 from the previous one.
Understanding this value helps in predicting any term or summing up the series. In our example, the common difference \(d\) is 3. This means each term increases by 3 from the previous one.
- To find the common difference, simply subtract the first term from the second term.
- It is a constant value that applies to all pairs of consecutive terms in the series.
Number of Terms
The number of terms in an arithmetic series dictates how many elements there are to sum up.
It is denoted by \(n\) in the sum formula. To find \(n\), use the last term and solve using the general term equation. For our series, we calculated \(n = 37\).
It is denoted by \(n\) in the sum formula. To find \(n\), use the last term and solve using the general term equation. For our series, we calculated \(n = 37\).
- Calculate the number of terms by rearranging the formula: \[ a_n = a + (n-1)d \]
- Substitute the known last term and solve for \(n\).
- The number of terms determines how the sum grows, as more terms mean a larger total.
