Problem 30 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=-4, a_{20}=15 $$

Step-by-Step Solution

Answer

The sum of the first 20 terms is 110.

Step 1: Identify the parameters

First, we identify the parameters given and needed for the sum of the first 20 terms of the arithmetic sequence. We have the first term \( a_1 = -4 \) and the twentieth term \( a_{20} = 15 \). We need to find the sum \( S_{20} \).

Step 2: Find the common difference

In any arithmetic sequence, the \( n \)-th term is given by the formula \( a_n = a_1 + (n-1) d \). Using this formula for \( a_{20} = 15 \), we have:\[ 15 = -4 + (20-1) d \]This simplifies to:\[ 15 = -4 + 19d \]Solving for \( d \), we add 4 to both sides:\[ 19 = 19d \]Then divide by 19:\[ d = 1 \].

Step 3: Use the sum formula for arithmetic sequence

The formula for the sum of the first \( n \) terms of an arithmetic sequence is given by:\[ S_n = \frac{n}{2} (a_1 + a_n) \]For the first 20 terms, \( n = 20 \), \( a_1 = -4 \), and \( a_{20} = 15 \). Substitute these values into the formula:\[ S_{20} = \frac{20}{2} (-4 + 15) \]This simplifies to:\[ S_{20} = 10 \times 11 = 110 \].

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence is a fundamental concept that allows us to find the total of a set of numbers joined by a consistent pattern. Understanding how to calculate this sum provides a powerful tool for solving various types of problems involving sequences.

In an arithmetic sequence, the sum of the first \( n \) terms is expressed using the formula:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

This formula helps in summing up the sequence by considering the first and the last terms. The "\( n \)" represents the number of terms, "\( a_1 \)" is the first term, and "\( a_n \)" is the last term in the series.

For example, let's calculate the sum of the first 20 terms in the sequence where \( a_1 = -4 \) and \( a_{20} = 15 \). Plugging into the formula:

• \( S_{20} = \frac{20}{2} (-4 + 15) \)

You find that:

• \( S_{20} = 10 \times 11 = 110 \)

It illustrates how each term's value incrementally adds to the total sum, building upon the consistent difference between terms.

Common Difference

In the world of arithmetic sequences, the common difference is key to understanding how sequences progress. This difference is a constant value added to each term to arrive at the next one, and it defines the sequence's nature.

Knowing how to find this difference is crucial, especially when you need to calculate terms further in the sequence or derive specific terms like the last one in the given sequence. The common difference, denoted by \( d \), can be determined by using any two consecutive terms or the formula:

• \( a_n = a_1 + (n-1)d \)

Here, we found \( d = 1 \) in the problem by substituting first and the twentieth terms into the equation:

• \( 15 = -4 + 19d \)

Solving it provides the difference:

• \( d = 1 \)

With this constant step of 1, each term increases steadily, making predicting future terms or their sums straightforward.

Arithmetic Sequence Formula

The arithmetic sequence formula is a crucial aspect of understanding patterns in sequences. It's the backbone for defining and locating specific terms in a sequence. This formula, \( a_n = a_1 + (n-1)d \), serves multiple purposes:

• It allows you to find any term within the sequence.
• It helps in identifying the sequence's common difference.
• It lays the groundwork for calculating the sequence's sum.

To apply the formula correctly, you need to know:

• \( a_1 \), the first term.
• \( n \), the term number you are interested in.
• \( d \), the common difference.

Using our example, where \( a_1 = -4 \) and \( d = 1 \), finding the twentieth term confirms the sequence's values:

• \( a_{20} = -4 + 19 \times 1 = 15 \)

The arithmetic sequence formula not only simplifies term calculation but also augments comprehension of the sequence's structure. It highlights how each term builds on the last, maintaining consistency across the series.