Problem 56 - Algebra 2

Question

Evaluate the series \(\sum_{n=1}^{40}\left(10-\frac{n}{2}\right)\) Show your work.

Step-by-Step Solution

Answer

The sum of the series is 380.

Step 1: Break down the series

Break down the series into two parts: \(\sum_{n=1}^{40} 10\) and \(-\sum_{n=1}^{40} \frac{n}{2}\) This is valid because the sum of sums is equal to the sum of the individual parts.

Step 2: Calculate the sum of the first part

Calculate the sum of the first part \(\sum_{n=1}^{40}10\) by multiplying 10 (the constant term) by 40 (number of terms). The result is 400.

Step 3: Calculate the sum of the second part

Calculate the sum of the second part \(-\sum_{n=1}^{40} \frac{n}{2}\) which is an arithmetic series with first term (\(a_{1}\)) 0.5, common difference (d) 0.5, and number of terms (n) 40. Use the formula for the sum \(S = \frac{n}{2} (a_{1}+a_{n})\), which simplifies to \(S = 20*1 = 20\). Remember to take the negative of the sum.

Step 4: Sum the results

Add the results of the first part (400) and the second part (-20) to get the final answer: 400-20 = 380.

Key Concepts

Summation Formula

When working with series, one of the most essential tools is the summation formula. It allows us to efficiently calculate the sum of a sequence of numbers. For arithmetic series where each term increases by a constant value, the formula is given by:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Here, \( S_n \) represents the sum of the series, \( n \) is the number of terms in the series, \( a_1 \) is the first term, and \( a_n \) is the last term. This formula helps to quickly find the sum without needing to add each term individually. In the context of our exercise, it was used to evaluate the arithmetic series portion where each term decreased by 0.5.

• Useful for simple and complex series calculations.
• Reduces time spent on manual calculations.

Understanding the summation formula is a fundamental skill in mathematics, especially in calculus and higher-level math, as it lays the groundwork for evaluating more complex series.

Constant Term in Series

Dealing with a constant term in any series simplifies the calculation process considerably. When each term in the series is the same constant value, it becomes straightforward to find the sum.
In the exercise, the series \( \sum_{n=1}^{40} 10 \) contains a constant term, 10. To find the sum, simply multiply this constant by the number of terms (40 in this case). The formula can be expressed as:
\[ \sum_{n=1}^{N} c = c \times N \]where \( c \) is the constant term, and \( N \) is the number of terms.
Here's why it's effective:

• Saves time since you don't need to add each value individually.
• Ensures accuracy by reducing the calculations to a simple multiplication.

By knowing and applying this principle, you can handle parts of the series efficiently and focus on more complex sections that may require additional analysis.

Calculating Partial Sums

Calculating partial sums involves breaking down a complex series into simpler parts. This makes it easier to manage and solve. In our exercise, two partial sums were considered separately:

• One part was the constant term series, \( \sum_{n=1}^{40} 10 \).
• The other part was the arithmetic sequence, \( -\sum_{n=1}^{40} \frac{n}{2} \).

By splitting the series into these parts, we direct our focus and simplify the computation. Calculating the sum of each segment individually before combining them is a strategic approach. After calculating each section, the results are simply added together to find the total sum of the series.This method is essential because:

• It offers a structured approach to complex problems.
• Allows for pinpointing errors, as each section is calculated separately.

Series Evaluation Methods

Choosing the right series evaluation method can greatly affect how efficiently we can solve a problem. Different strategies can be applied depending on the series structure:

• Direct Calculation: As seen in constant terms, where a direct multiplication suffices.
• Use of Formulas: Arithmetic formulas, like the summation formula, provide a shortcut to find sums without calculating each term individually.
• Breaking Down Complex Series: Dividing a series into parts, known as partial sums, to evaluate them separately can simplify calculations.

In the given exercise: - Direct calculation was employed for the constant term. - The summation formula helped in handling the arithmetic series. - Combining results of different parts gave the final answer efficiently. Bear in mind that understanding and applying appropriate series evaluation methods are crucial skills, enabling not just accuracy but also enhancing your problem-solving speed.