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Chapter 11: Problem 68

Solve each problem. Integer Sum Find the sum of all the integers from \(-8\) to 30 .

Short Answer

Expert verified
The sum of all integers from -8 to 30 is 429.

Step by step solution

01

- Identify the Sequence

Recognize that the problem involves finding the sum of an arithmetic sequence from -8d to 30. Terms are consecutive integers, hence the common difference (d) is 1.
02

- Calculate the Number of Terms

Use the formula for the nth term of an arithmetic sequence: [ a_n = a + (n-1)d ]. We set the last term to 30, the first term to -8, and solve for n.\[ 30 = -8 + (n-1) \cdot 1 \]Solve for n:\[ 30 + 8 = n-1 \]\[ 38 = n-1 \]\[ n = 39 \]So, there are 39 terms.
03

- Use the Sum Formula for Arithmetic Series

The sum S_n of an arithmetic sequence can be calculated with: \[ S_n = \frac{n}{2} \cdot (a + l) \]Here,\[n = 39, a = -8, l = 30\]Substitute into the formula:\[ S_{39} = \frac{39}{2} \cdot (-8 + 30) \]\[= \frac{39}{2} \cdot 22 \]\[= 39 \cdot 11 \]\[= 429 \]Therefore, the sum of all integers from -8 to 30 is 429.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integer sum
An integer sum refers to the process of adding together whole numbers, both positive and negative. **In the exercise given**, we need to calculate the sum of all integers from \(-8\) to 30. The sequence here is straightforward since it consists of consecutive numbers adding one after the other. To find this sum accurately, it involves understanding and applying the concept of arithmetic sequences and using the correct formula. **Key to solving this is realizing:**
- The range of numbers forms an arithmetic sequence.
- Using the correct number of terms ensures your formula accurately computes the sum. The beauty of integers is in their simplicity; there's no need to worry about fractions or decimals here - just clean, whole numbers.
arithmetic series
An arithmetic series is simply the sum of the numbers in an arithmetic sequence. An arithmetic sequence is a list of numbers where the difference between consecutive terms, called the common difference (\reinterpret{d}), is the same. **In this problem**, the sequence is from \(-8\) to 30. Here, the common difference \(d\) is 1 because each number increments by 1.

To find the number of terms (**\(n\)**) in the series, we use the formula for the nth term of an arithmetic sequence:
\

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Most popular questions from this chapter

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