Problem 50 - College Algebra

Question

Write out the first three terms and the last term. Then use the formula for the sum of the first \(n\) terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{50}-4 i$$

Step-by-Step Solution

Answer

The first three terms of the sequence are -4, -8, -12. The 50th term is -200. The sum of the first 50 terms is -2020.

Step 1: Find the First Three Terms and Last Term

The first three terms of the sequence are -4, -8, and -12. These numbers are obtained from the series \( \sum_{i=1}^{50}-4 i \) which in general is a sequence of the form -4i with i ranging from 1 to 50. Similarly, to find the last (50th) term of the arithmetic sequence, use the formula \( a + (n - 1) * d \), which with a=-4, n=50, and d=-4 gives -200.

Step 2: Find Sum of the First 50 Terms

Next, use the formula for the sum of an arithmetic series \(\frac{n}{2} * (a_1 + a_n)\), where \(a_1 = -4\) is the first term, \(a_n = -200\) is the last term, and \(n = 50\) is the number of terms. Substituting these values into the formula gives \(-2020\). Thus, the sum of the first 50 terms in this arithmetic sequence is \(-2020\).

Key Concepts

Sum of Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. To calculate this, you can use a specific formula that simplifies the process. This is especially handy when you have a large number of terms, like 50, and don't want to add each one individually.
The formula for finding the sum of an arithmetic series is \[ S_n = \frac{n}{2} \times (a_1 + a_n) \], where:

• \( S_n \) is the sum of the series.
• \( n \) is the number of terms.
• \( a_1 \) is the first term.
• \( a_n \) is the last term.

By using this formula, you can quickly compute the sum by just knowing the first and the last terms. For example, if you have 50 terms, and they start at -4 and end at -200, you plug those into the formula: \[ S_{50} = \frac{50}{2} \times (-4 + (-200)) = -2020 \].
This approach simplifies calculating the sum of many terms and is crucial for solving complex arithmetic problems effectively.

Sequence Terms

Sequence terms are the individual elements in a sequence. In an arithmetic sequence, these terms follow a specific pattern established by the first term and the common difference. Each term is created by adding the common difference to the previous term.
For instance, let's consider the sequence \(-4, -8, -12, \ldots, \). This sequence starts at -4 and each subsequent term is 4 less than the previous one because the common difference is -4.
Understanding sequence terms is essential because it helps in identifying the properties of the sequence such as how it grows or shrinks, and it lays the foundation for using formulas related to arithmetic sequences.

Formula for Arithmetic Sequence

The formula for finding any term in an arithmetic sequence is an essential tool for solving sequence-related problems. This formula is \( a_n = a_1 + (n - 1) \times d \), where:

• \( a_n \) is the term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference between consecutive terms.

Using this formula allows you to find any term without having to list all the previous terms. For example, in the sequence \(-4, -8, -12, \ldots\), if you want to find the 50th term, you can use this formula. Given \( a_1 = -4 \), \( n = 50 \), and \( d = -4 \), you would substitute these into the formula to get \( a_{50} = -4 + (50 - 1) \times (-4) = -200 \).
This makes it easy to determine any term in a sequence, saving time and effort in calculations.

Finding Sequence Terms

Finding sequence terms is about identifying the values within a sequence using a general rule or formula. This often involves determining a first few terms quickly to understand the pattern.
In arithmetic sequences, finding terms is straightforward once you have the first term and the common difference. For example, starting with -4 and using a common difference of -4, the first three terms can be calculated as:

• First term, \( a_1 = -4 \).
• Second term, \( a_2 = -4 + (-4) = -8 \).
• Third term, \( a_3 = -8 + (-4) = -12 \).

This straightforward approach helps establish the structure of the sequence and aids in using different formulas for further calculations like finding the sum of an arithmetic series.