Problem 25 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=4, d=2 $$

Step-by-Step Solution

Answer

The sum of the first 20 terms is 460.

Step 1: Identify the Formula

For an arithmetic sequence, the sum of the first \( n \) terms, denoted as \( S_n \), is given by the formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \] where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

Step 2: Substitute the Known Values

Start by substituting the given values into the formula. We have \( a_1 = 4 \), \( d = 2 \), and \( n = 20 \). The formula becomes: \[ S_{20} = \frac{20}{2} \times (2 \times 4 + (20-1) \times 2) \]

Step 3: Perform the Calculations Inside the Parentheses

Calculate the terms inside the parentheses: \( 2 \times 4 = 8 \) and \((20-1) \times 2 = 38\). The expression inside the parentheses becomes \( 8 + 38 = 46 \).

Step 4: Calculate the Sum

Now calculate the entire expression: \[ S_{20} = 10 \times 46 = 460 \] Thus the sum of the first 20 terms is 460.

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence can be easily calculated using a specific formula. This formula is key because it allows you to add up all the terms in the sequence quickly. Imagine having to add each number individually—this formula saves you a lot of time!
To find the sum of the first \( n \) terms in an arithmetic sequence, you use:

• \( S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \)

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

This formula works because it essentially finds the average of the first and last term and multiplies it by the number of terms. This way, you can quickly calculate the sum for sequences of any length without error, like when we found the sum to be 460 for 20 terms.

First Term

The first term, often denoted as \( a_1 \), is crucial in starting off an arithmetic sequence. Think of it as the foundation. Just like the start line in a race, \( a_1 \) tells you where the sequence begins and helps determine the value of every other term.
In our exercise, we were given that \( a_1 = 4 \). This means that the sequence starts at 4. Using \( a_1 \) is important because it plugs directly into the sum formula, helping to determine what every term will be as it continues from this starting point. Without knowing the first term, it's impossible to accurately calculate either individual term values or the full sequence sum.

Common Difference

The common difference, denoted as \( d \), is what sets arithmetic sequences apart from other types of sequences. It is the amount by which each term in the sequence increases or decreases compared to the previous one.
In our example, the common difference was \( d = 2 \). This means each number in the sequence is 2 more than the number before it. So starting from the first term, \( 4 \), the sequence goes 4, 6, 8, 10, and so on. This consistent difference allows us to predict and calculate not just the terms within the sequence, but also helps in using our sum formula effectively.
By knowing \( d \), you can foresee the progression of the sequence and correctly complete formulas that depend on this information. It's like having a clear map or a navigational guide for your arithmetic sequence journey.