Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of all the integers from \(-8\) to 30.

Step-by-Step Solution

Answer

The sum is 429.

Step 1: Determine the Number of Terms

First, identify the sequence of integers from \(-8\) to 30. This sequence is arithmetic with the first term \(a_1 = -8\) and the last term \(a_n = 30\). The common difference \(d = 1\). We need to find \(n\), the number of terms.Use the formula for the \(n\)-th term of an arithmetic sequence: \[a_n = a_1 + (n-1) \cdot d\]Set \(a_n = 30\), \(-8 = a_1\), and \(d = 1\):\[30 = -8 + (n-1) \cdot 1\]Solve for \(n\):\[30 = -8 + n - 1\] \[n = 30 + 8 + 1 = 39\]

Step 2: Calculate the Sum of the Arithmetic Sequence

Now, use the formula for the sum \(S_n\) of an arithmetic sequence:\[S_n = \frac{n}{2} \cdot (a_1 + a_n)\]Substitute \(n = 39\), \(a_1 = -8\), and \(a_n = 30\) into the formula:\[S_{39} = \frac{39}{2} \cdot (-8 + 30)\]Calculate the terms inside the parentheses and then the entire expression:\[S_{39} = \frac{39}{2} \cdot 22 = \frac{39 \cdot 22}{2} = \frac{858}{2} = 429\]

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence is found by adding up all the terms between the first and last term in the sequence. This can be done efficiently using a special formula. For arithmetic sequences, the sum can be calculated without having to add each term individually, which saves a lot of time and effort, especially for longer sequences.
The formula used is:

• \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\)

Here, \(S_n\) is the sum of the sequence, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term in the sequence.
This formula works because it pairs the terms from the sequence symmetrically around the middle, making it easy to compute the sum by leveraging these pairs.

Number of Terms in Sequence

To find the sum of an arithmetic sequence, you first need to determine how many terms, \(n\), there are in the sequence. This involves understanding the basic structure of the sequence.
Given the formula:

• \(a_n = a_1 + (n-1) \cdot d\)

where \(a_n\) is the last term, \(a_1\) is the first term, and \(d\) is the common difference, you can rearrange this formula to find \(n\):

• \(n = \frac{a_n - a_1}{d} + 1\)

Understanding this formula helps you count all the terms in the sequence from start to finish.
Once you have \(n\), you can then proceed to calculate other properties of the sequence, like its sum.

Common Difference

In any arithmetic sequence, the common difference \(d\) is the fixed amount added to each term to get to the next term. It's an essential characteristic that defines the progression of the sequence, ensuring each term follows consistently from the last.
Determining the common difference is straightforward:

• Subtract any term from the one that follows it.

For example, if you have the sequence: \(2, 4, 6, 8, ...\), the common difference \(d\) is 2, since each number is 2 more than the previous.
Knowing \(d\) is crucial as it enables you to use formulas to find the number of terms or the total of the sequence, integrating smoothly into calculations like finding individual terms \(a_n\) or the overall sum \(S_n\).