Problem 38 - A Graphical Approach to Precalculus with Limits

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 of the first 10 terms is 90.

Step 1: Understand the problem

To find the sum of the first 10 terms of an arithmetic sequence with the first term \(a_1 = -9\) and common difference \(d = 4\), we can use the formula for the sum of the first \(n\) terms of an arithmetic sequence: \[ S_n = \frac{n}{2} (2a_1 + (n-1) d) \]. This formula allows us to calculate the sum directly by substituting the values for \(n\), \(a_1\), and \(d\).

Step 2: Substitute known values

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

Step 3: Calculate the expression inside the parentheses

First, calculate \(2(-9)\): \[ 2(-9) = -18 \]. Next, calculate \((10-1) \cdot 4\): \[ 9 \cdot 4 = 36 \]. Then, combine the results: \[ -18 + 36 = 18 \].

Step 4: Calculate the sum

Now compute \(S_{10}\) using the simplified expression: \[ S_{10} = 5 \cdot 18 = 90 \]. This is the sum of the first 10 terms of the arithmetic sequence.

Key Concepts

Sum of Series

The sum of an arithmetic series is a way to add up several numbers in a sequence that has a common difference. When you hear 'arithmetic sequence,' think of a pattern of numbers that increase or decrease by the same amount each time. To find the sum of such a sequence, you use the formula:\[S_n = \frac{n}{2} (2a_1 + (n-1) d)\]Where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms.
• \( a_1 \) is the first term.
• \( d \) is the common difference.

So, if you know these three things, you can use this formula to quickly find the total sum of a particular part of the sequence. This simplifies calculations and helps avoid any mistakes.
The beauty of this formula is how it compacts all repetitive additions into one simple calculation.

Common Difference

The common difference \(d\) in an arithmetic sequence is the amount each term increases or decreases from the previous one. It's a constant value that keeps the sequence evenly spaced.
Consider the example where the first term \(a_1\) is \(-9\) and \(d = 4\). Here’s how it works:

• The sequence starts at \(-9\).
• Each subsequent term is found by adding \(4\) to the last term.

So, the first few terms would be: \(-9, \ -5, \ -1, \ 3, \ 7, \ 11,\ 15, \ 19, \ 23, \ 27\). You can clearly see each number is four greater than the one before it.
The common difference tells you exactly how the sequence changes and grows. Knowing this helps you predict future terms without calculation.

First Term

The first term \(a_1\) is where every arithmetic sequence begins. It's like the starting point of a journey. In our example series, the first term is \(-9\), and it serves as the anchor for everything that follows.
Understanding the first term is essential for applying the sum of the series formula or any analysis of the sequence. It's the point from which every calculation stems.

• If the first term is negative, like \(-9\), it means the sequence starts below zero.
• If positive, it starts above zero.

The first term, when combined with the common difference, defines the entire series. So, grasping this concept helps in both constructing and dissecting arithmetic sequences.