Problem 48 - College Algebra

Question

For Exercises \(45-50,\) 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. $$\sum_{i=1}^{40}(-2 i+6)$$

Step-by-Step Solution

Answer

The first three terms of the sequence are 4, 2, 0, and the last term is -74. The sum of the sequence is -1400.

Step 1: Identify the first three terms of the sequence

To identify the first three terms, substitute \(i=1,2,3\) respectively into \(-2i + 6\) that result as first term \(4\), second term \(2\), third term \(0\)

Step 2: Calculate the last term

The last term of the sequence can be calculated by substituting the last term number \(i=40\) into the term formula \(-2i + 6\), which gives \(-2*40+6=-74\)

Step 3: Calculate the sum of the series

Utilising the formula for the sum of an arithmetic sequence, \(S = n/2 * (a + l)\), where \(n = 40\), \(a = 4\), and \(l = -74\), the result for \(S\) is calculated as \(S = 40/2 * (4 - 74) = -1400\)

Key Concepts

Sum of arithmetic sequence

In mathematics, an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. When we want to find the total of all terms in such a sequence, we refer to this as the sum of an arithmetic sequence. This can be a handy calculation, especially when dealing with a large number of terms.
To calculate the sum, we often use the formula for the sum of the first n terms:

• Formula: \( S = \frac{n}{2} \times (a + l) \)
• Where: \( S \) is the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

This approach allows us to calculate the sum without having to manually add each term, making it computationally efficient. For example, if we want to find the sum of the first 40 terms of the sequence defined by \(-2i + 6\), after calculating the first term (4) and the last term (-74), we find that the sum \( S \) is \(-1400\).

First and last terms

The first and last terms in an arithmetic sequence serve as the basis for calculating the sum of the sequence. Identifying these terms is crucial because they directly feed into the sum formula.

• First Term: For the sequence formula \(-2i + 6\), begin by substituting the first term's position, \(i = 1\). This gives us \(-2 \times 1 + 6 = 4\).
• Last Term: Similarly, substitute the last term's position, \(i = 40\), into the sequence formula. This results in \(-2 \times 40 + 6 = -74\).

These terms eventually help determine the total sum of the sequence, along with the number of terms \( n \). This approach highlights the importance of understanding how the sequence is constructed and how simple substitutions can give us the terms needed for further calculations.

Sequence formula evaluation

Evaluating a sequence formula involves understanding how to derive the terms of the sequence mathematically. The given sequence formula \(-2i + 6\) is an example of a linear expression used to generate terms based on the variable \(i\).

• Substitution: For any specific term, substitute the respective term number into the formula. For example, for the third term where \(i = 3\), we substitute to get \(-2 \times 3 + 6 = 0\).
• Understanding the Pattern: The formula tells us that each term changes by a constant difference. Here, every subsequent term decreases by 2, evident from the coefficient of \(i\), which is \(-2\). Therefore, it offers insight into the sequence's progression.

This process of formula evaluation helps in efficiently finding any term within the sequence, which in turn expedites solving problems like finding the sum of the sequence. Understanding how the formula works takes the guesswork out of determining specific terms, enabling precise calculations.