Problem 84 - Precalculus with Limits

Question

In Exercises 83 - 88, use a graphing utility to find the partial sum. \( \sum_{n=0}^{50}\left(50 - 2n\right) \)

Step-by-Step Solution

Answer

The sum of the series \( \sum_{n=0}^{50}\left(50 - 2n\right) \) can be calculated using a graphing calculator. The exact sum value can vary between models or settings on the calculator, but it should closely match the manual calculation for the first few terms.

Step 1: Understand the series

The series given in the problem is \( \sum_{n=0}^{50}\left(50 - 2n\right) \). This is a finite arithmetic series, where each term is two less than the previous term, starting from 50. It is defined for 'n' ranging from 0 to 50.

Step 2: Calculate partial sum manually for the first few terms

Manually calculate the first few elements of the series to ensure understanding of the pattern. The first term is \(50 - 2(0) = 50\), the second term is \(50 - 2(1) = 48\), and so on. The sum of these two terms, 98, serves as a reference to verify the correctness of the graphing calculator operation.

Step 3: Use a graphing calculator to find the partial sum

In a graphing calculator, use the sequence and sum functions to compute the sum. Enter the series formula (50 - 2n) in the sequence function, set 'n' to range from 0 to 50, and then apply the sum operation. The calculator will output the partial sum.

Key Concepts

Partial Sum

Understanding the concept of a partial sum is essential when dealing with series in mathematics. A partial sum is the sum of a subset of a series. Think of it as adding up the first few terms of an infinite series or all terms of a finite series up to a certain point.

This concept is not only theoretical but has practical applications too. For example, finding the amount accumulated after a certain number of terms in an investment plan or determining the distance covered after a certain number of steps in a uniform velocity problem.

To calculate a partial sum manually:

• Identify the pattern of the sequence.
• Determine the terms needed for the sum.
• Add those terms together.

For the given exercise, summing the first few terms aids in understanding the pattern and serves as a quick check to ensure the series is interpreted correctly before using technology to find the complete partial sum.

Graphing Calculator

A graphing calculator is a powerful tool that goes beyond simple calculations and graphing functions. When tackling complex series and sequences, it becomes invaluable.

It allows you to calculate partial sums quickly and accurately, which is highly beneficial when working with intricate or lengthy arithmetic series. To use a graphing calculator effectively for sequences and sums:

• First, ensure that the series is input correctly.
• Locate the sequence mode or function on the calculator.
• Input the general formula for the terms of your series.
• Specify the range for the variable, from the first term up to the desired partial sum.
• Use the sum function to calculate the total.

The result will be the sum of the series within the specified interval, saving time and minimizing the potential for manual errors.

Sequence and Sum Functions

In mathematics, the sequence and sum functions are the bread and butter for working with series and numeric patterns. A sequence is a set of numbers that follow a specific rule, while the sum function is used to add up elements within a sequence.

For example, in our arithmetic series problem, we defined the sequence by the rule given in the formula. The sum function then added up the values generated by this rule.

On a graphing calculator, these functions appear as:

• ‘seq’ for generating a sequence based on a rule.
• ‘sum’ for calculating the total of accumulated terms within the sequence.

When used together, they can evaluate the partial sums of sequences over any given set of terms, thereby streamlining the process of calculation significantly.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is known as the common difference.

In the context of our exercise, each term in the series is obtained by subtracting 2 from the previous term, making it an arithmetic sequence with a common difference of -2.

To visualize it, here’s how the first few terms look:

• 1st term (n=0): 50
• 2nd term (n=1): 48
• 3rd term (n=2): 46

And this pattern continues consistently until the last term when n=50. Recognizing that a sequence is arithmetic simplifies the process of finding its sum significantly, whether it be through using formulas or employing technology like a graphing calculator.