Problem 34 - A Graphical Approach to College Algebra

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=-9, d=4$$

Step-by-Step Solution

Answer

The sum is 90.

Step 1: Understand the Problem

We need to find the sum of the first 10 terms of an arithmetic sequence where the first term \( a_1 = -9 \) and the common difference \( d = 4 \). This requires finding each term of the sequence and then adding them together.

Step 2: Use the Formula for the Sum of an Arithmetic Sequence

The sum of the first \( n \) terms of an arithmetic sequence can be calculated using the formula: \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]where \( S_n \) is the sum, \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 3: Substitute Known Values into the Formula

In this case, \( n = 10 \), \( a_1 = -9 \), and \( d = 4 \). Substitute these values into the sum formula:\[ S_{10} = \frac{10}{2} (2(-9) + (10-1)4) \]

Step 4: Perform the Calculations

First, calculate the expression inside the parentheses:- Calculate \( 2(-9) = -18 \).- Calculate \((10-1)4 = 36\).- Sum inside the parentheses: \(-18 + 36 = 18\).

Step 5: Finalize the Calculation

Now, substitute back to determine \( S_{10} \):\[ S_{10} = \frac{10}{2} \times 18 \]\[ S_{10} = 5 \times 18 \]\[ S_{10} = 90 \]

Step 6: Conclusion

The sum of the first 10 terms is \( 90 \).

Key Concepts

Sum of Series

Finding the sum of an arithmetic series is like adding a large set of numbers efficiently. Instead of adding each number individually, mathematics provides us a neat formula. For arithmetic series, the sum of the first \( n \) terms (_sum of series_), \( S_n \), can be calculated using:\[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \].Let's make sense of each part:

• \( n \) is the total number of terms we want to sum.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference, or the amount we add to each term to get the next.

This formula gives us a shortcut. Instead of computing each term and then adding them, we can quickly find the sum. Plug in our values to get the result like a multiplication problem - fast and accurate!

Common Difference

The concept of _common difference_ is the heart of arithmetic sequences. It's simply how much we add (or subtract) to get from one term to the next. If the common difference \( d \) is positive, the sequence increases. If negative, it decreases.Let's see how this works using our example:

• First term \( a_1 = -9 \)
• Common difference \( d = 4 \)

To find the next term, add \( 4 \) (our common difference) to \( -9 \). This gives \( -5 \), and so on. Continuing this pattern, you'll spot that the terms are like stepping stones that are always \( 4 \) units apart. It's like walking up a staircase of equal steps.

Number of Terms

Understanding the _number of terms_ \( n \) in a sequence is simple but crucial. In any list of values, the number of terms tells you how many numbers are in the list. For our arithmetic sequence, \( n = 10 \) indicates we are considering the first 10 numbers from the sequence.This number is key for calculating the sum. Why? Because it's used directly in the sum formula. Think of each term as a piece in a puzzle, with \( n \) being the count of these puzzle pieces. In arithmetic sequences, this number also helps determine the last term in the sequence, which provides boundaries for finding the sum. Just remember, in an arithmetic sequence like ours, these terms are equally spaced out, thanks to the common difference.