Problem 28 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=0, d=-4 $$

Step-by-Step Solution

Answer

The sum of the first 20 terms is -760.

Step 1: Identify the Formula

To find the sum of the first \( n \) terms of an arithmetic sequence, we use the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \) where \( a_1 \) is the first term and \( a_n \) is the nth term.

Step 2: Calculate the Nth Term

First, we need to find the 20th term of the sequence, \( a_{20} \), using the formula: \( a_n = a_1 + (n-1) \times d \). Here, \( a_1 = 0 \), \( d = -4 \), and \( n = 20 \). Substitute these into the formula:\[a_{20} = 0 + (20-1) \times (-4) = 0 - 76 = -76\]

Step 3: Substitute Values into the Sum Formula

Now that we have \( a_1 = 0 \) and \( a_{20} = -76 \), we can calculate the sum of the first 20 terms using the sum formula:\[S_{20} = \frac{20}{2} \times (0 + (-76)) = 10 \times (-76) = -760\]

Step 4: Finalize the Calculation

The sum of the first 20 terms of the sequence has been calculated as \(-760\). Thus, the sum of the sequence is \(-760\).

Key Concepts

Understanding the Sum of an Arithmetic Sequence

The sum of an arithmetic sequence, often referred to as an arithmetic series, tells us the total of all terms from the first up to a certain term. For any arithmetic sequence, there's a quick and handy formula to find this sum, especially useful when the number of terms becomes large. The formula \[ S_n = \frac{n}{2} (a_1 + a_n) \] allows you to find the sum without having to manually add all terms together.

• \( S_n \) stands for the sum of the first \( n \) terms.
• \( a_1 \) is the first term of the sequence.
• \( a_n \) is the \( n \)-th term, or the last term we are adding.
• \( n \) is the total number of terms we are summing.

Using this formula simplifies the process significantly and helps in finding the sum efficiently. This is especially helpful in long sequences, like finding the sum of the first 20 terms in our example.

Exploring the Nth Term Formula of an Arithmetic Sequence

To work with any arithmetic sequence, knowing how to find any term in the sequence is essential. The formula for the nth term is straightforward: \[ a_n = a_1 + (n-1) \times d \] Here,

• \( a_n \) represents the \( n \)-th term of the sequence.
• \( a_1 \) is again the first term.
• \( d \) represents the common difference, which is the amount adding to each term to get to the next.
• \( n \) is the term number you wish to find.

This formula is crucial because it allows you to find any term's value without listing all the previous terms. For instance, to find the 20th term of our sequence with \( a_1 = 0 \) and \( d = -4 \), we simply substitute into the formula and solve to get \( a_{20} = -76 \). This value then helps us in calculating the sequence's sum.

Understanding the Concept of the Common Difference

At the heart of any arithmetic sequence is the common difference. This is the fixed number we're adding (or subtracting) to each term to generate the next term. It can be positive, making the sequence increase, or negative, making it decrease, as in our example where \( d = -4 \).

• Having a common difference of \( d \) means every term changes by \( d \) from the previous one.
• If \( d \) is negative, every step subtracts that value, leading the sequence to decrease.
• The common difference plays a role in the nth term formula \( a_n = a_1 + (n-1) \times d \), affecting every term's value directly.

Understanding and identifying this common difference is central to solving problems involving arithmetic sequences. It influences both the specific terms and the overall sum, acting as a control for the sequence's behavior.