Problem 47 - Algebra for College Students

Question

(-7)+(-10)+(-13)+(-16)+\cdots+(-109) \quad-2030 \quad=

Step-by-Step Solution

Answer

The result of the expression is -4060.

Step 1: Recognize the Sequence

The sequence provided is arithmetic with a common difference. The sequence begins with -7, and each subsequent term decreases by 3, i.e., -7, -10, -13, and so on.The general form for the nth term (a_n) in an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) imes d \]where:- \(a_1\) is the first term (-7 in this case),- \(d\) is the common difference (-3 in this case).

Step 2: Determine the Number of Terms

We need to find how many terms there are until the last term -109. Using the formula for the nth term:\[ a_n = a_1 + (n-1) imes d \]Substitute the known values:\[ -109 = -7 + (n-1) imes (-3) \]Solve for \(n\):\[ -109 = -7 - 3(n-1) \]\[ -109 + 7 = -3(n-1) \]\[ -102 = -3(n-1) \]\[ 102 = 3(n-1) \]\[ 34 = n-1 \]\[ n = 35 \]Thus, there are 35 terms in the sequence.

Step 3: Apply the Sum Formula for Arithmetic Sequences

The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence can be calculated using the formula:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]Substituting the values we found:\[ S_{35} = \frac{35}{2} \times (-7 + (-109)) \]\[ S_{35} = \frac{35}{2} \times (-116) \]\[ S_{35} = 35 \times (-58) \]\[ S_{35} = -2030 \]

Step 4: Solve for the Expression

The expression given was:\[ S_{35} - 2030 = 0 \]We found that \( S_{35} = -2030 \), thus:\[ -2030 - 2030 \]\[ = -4060 \]

Key Concepts

nth term formula

The nth term formula for an arithmetic sequence helps you find any term in the sequence without listing all the terms one by one. In an arithmetic sequence, each term increases or decreases by a constant amount, known as the common difference.
To determine the nth term, use the formula: \[ a_n = a_1 + (n-1) \times d \]

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

In the given exercise, you see that you start with \( a_1 = -7 \). Each term is decreasing by \( d = -3 \). To find how many terms go up to \( -109 \), you substitute into the nth term formula and solve for \( n \).
By setting up the equation \( -109 = -7 + (n-1) \times (-3) \) and solving, it becomes clear that \( n = 35 \) for the last term \( -109 \). You have successfully derived the position of the last term in the series.

common difference

The common difference in an arithmetic sequence is what separates it from other sequences—it is the consistent amount that each term changes from one to the next.
Consider the sequence in the problem: \( -7, -10, -13, -16, ... \)
This sequence decreases by 3 each time you move from one term to the next.

• Start with any two consecutive terms, for example, \(-10 - (-7) = -3 \).
• This calculation shows that \(-3\) is the common difference, noted as \(d\).

Recognizing this pattern allows you to apply the nth term formula more confidently to explore the sequence further. Whether finding a specific term or deriving a sum, the common difference \(d = -3\) is crucial for accurately interpreting the arithmetic progression.

sum of arithmetic sequence

The sum of an arithmetic sequence is the total you get when adding up all the terms in the sequence. This can be important for scenarios where you need the cumulative value of these numbers.
For an arithmetic sequence, you can find the sum of its first \( n \) terms with this formula:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \] Here's a breakdown:

• \( S_n \) represents the sum of the first \( n \) terms.
• \( n \) is the total number of terms you're summing.
• \( a_1 \) is the first term in the series.
• \( a_n \) is the nth term or last term in your series.

For instance, in this exercise, where you need to find the sum from \( -7 \) to \( -109 \), you find \( n = 35 \), \( a_1 = -7 \), and \( a_n = -109 \). Substituting these into the formula, you get:\[ S_{35} = \frac{35}{2} \times (-7 + (-109)) \]Calculating further gives you the sum \( S_{35} = -2030 \). This shows the entire total of all 35 terms in the sequence. Understanding this sum formula completes the examination of the arithmetic sequence tasks in the exercise.