Problem 38 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. Find the sum of the first 50 terms of the arithmetic sequence: \(-15,-9,-3,3, \dots\)

Step-by-Step Solution

Answer

The general formula for the nth term of the arithmetic sequence is \(a_{n} = -15 + (n - 1) * 6\), the 20th term of the sequence is 99, and the sum of the first 50 terms of the sequence is 6600.

Step 1: Determine the common difference

In an arithmetic sequence, the common difference \(d\) is obtained by subtracting a term from the term that follows it. From the given sequence: \(-9 - (-15)\) = 6. So, the common difference \(d\) is 6.

Step 2: Write the formula for the nth term

Using the formula for an arithmetic sequence \(a_{n} = a_{1} + (n - 1) * d\), and replacing \(a_{1}\) with -15 (the first term of the sequence) and \(d\) with 6, the formula for the nth term of this arithmetic sequence becomes \(a_{n} = -15 + (n - 1) * 6\).

Step 3: Find the 20th term

To find the 20th term, substitute \(n = 20\) in the formula. \(a_{20} = -15 + (20 - 1) * 6 = -15 + 114 = 99\). Therefore, the 20th term of the sequence is 99.

Step 4: Find the sum of the first 50 terms

The sum \(S_{n}\) of the first n terms of an arithmetic sequence can be found using the formula \(S_{n} = n/2 * (a_{1} + a_{n})\). In this case, the 50th term needs to be calculated first: \(a_{50} = -15 + (50 - 1) * 6 = -15 + 294 = 279\). Therefore, the sum of the first 50 terms is \(S_{50} = 50/2 * (-15 + 279) = 25 * 264 = 6600\).

Key Concepts

Nth Term Formula

Understanding the nth term formula is crucial for handling arithmetic sequences effectively. It's like finding a specific address in a long street—it gives you direct access to any term in the sequence. In essence, the nth term formula expresses the value of any term based on its position in the sequence.

The standard nth term formula for an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference between terms. This formula is invaluable because it eliminates the need to list out the entire sequence to find a specific term.

Arithmetic Series Sum

The arithmetic series sum reflects the total of all terms in an arithmetic sequence up to a certain point. Imagine you are stacking blocks—one for each term—and you want the total height of the stack. This is what the arithmetic series sum calculates.

To find the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence, you can use the formula \( S_n = n/2 * (a_1 + a_n) \). This formula effectively cuts the work in half by calculating the sum of the first and last term and multiplying by the number of terms divided by two. It's a shortcut that keeps you from having to add up every term individually, saving both time and effort.

Common Difference

The common difference, denoted as \( d \), is what makes an arithmetic sequence 'arithmetic.' It's the consistent gap between consecutive terms—much like the uniform step height in a staircase. You can find it by subtracting any term from the next one in the sequence.

For the sequence given, we calculate the common difference as \( -9 - (-15) = 6 \), indicating that each term increases by 6 from the previous term. This common difference is the backbone of the sequence, as it remains constant throughout and dictates the value of subsequent terms.

Sequence and Series

The terms sequence and series are often used interchangeably, but they represent different concepts. A sequence is like a list of numbers in a specific order, while a series is about accumulation—the sum of terms in a sequence.

Understanding the distinction is important. A sequence is defined by the formula for its terms, and a series is defined by the sum of those terms. When dealing with sequences, we're focused on identifying patterns and finding individual terms, but with series, our goal shifts to evaluating total sums or means, like finding the collective weight of a number of objects.