Problem 87 - Precalculus with Limits

Question

In Exercises 83 - 88, use a graphing utility to find the partial sum. \( \sum_{i=1}^{60}\left(250 - \dfrac{2}{5}i\right) \)

Step-by-Step Solution

Answer

The partial sum can be calculated using a graphing utility or manually by calculating each term of the series from i=1 to 60 and summing them up. The exact sum would depend on the calculations done using the series expression \( \sum_{i=1}^{60}\left(250 - \dfrac{2}{5}i\right) \).

Step 1 Understand the series

The series given is \( \sum_{i=1}^{60}\left(250 - \dfrac{2}{5}i\right) \) which means summing the terms of the series from i = 1 to 60

Step 2 Calculate each term of the series

The terms of the series can be calculated by substituting values from 1 to 60 in the series formula and calculating each term. For instance, the first term is calculated by substituting i=1 in the expression, \( T_i = 250 - \dfrac{2}{5}i \) to get \( T_1 = 250 - \dfrac{2}{5}*1 = 249.6 \)

Step 3 Calculate the partial sum

The partial sum is the sum of all the terms from i=1 to 60. So calculate each term as in step 2 and add them up. Either this can be done manually or using a graphing utility as instructed by the problem.

Key Concepts

Partial Sum

A partial sum refers to the sum of a specific number of consecutive terms in a series. In the given exercise, the aim is to find the partial sum of the series starting from the first term up to the 60th term.

• This involves calculating each individual term in the series using the provided series formula: \(T_i = 250 - \dfrac{2}{5}i\).
• For example, the first term \(T_1\) is calculated by setting \(i\) to 1 in the formula, yielding \(T_1 = 249.6\).
• Continuing this process for all terms up to \(T_{60}\), and then summing these terms together, gives you the required partial sum.

While performing each calculation manually might illustrate the concept clearly, make sure to follow the instructions or use tools like a graphing utility to save on effort and time.

Graphing Utility

A graphing utility is a tool, often found in calculators or computer software, designed to help visualize mathematical equations and perform complex calculations with ease. In the context of this exercise:

• A graphing utility can effortlessly calculate the sum of numerous terms in a series by executing a sum function that iterates through the series formula over specified bounds \(i=1\) to \(i=60\).
• By inputting the series formula directly into the utility, it performs the calculations for each term, sums them, and provides you with the final result.
• This not only minimizes the likelihood of manual errors but also saves significant time compared to manual calculations.

Utilize graphing utilities whenever applicable, especially for large or complex calculations in order to increase efficiency and accuracy.

Series Formula

The series formula is a mathematical expression used to describe the pattern of terms in the sequence you are summing. In this exercise, the series formula is \(T_i = 250 - \dfrac{2}{5}i\), which shows each term is generated by a linear relationship based on the variable \(i\).

• The series formula is crucial because it defines the rule or relationship among consecutive terms in the series.
• Understanding how \(T_i\) changes with each \(i\) helps in predicting the behavior of the series and thus assists in calculating each term.
• In this case, as \(i\) increases, the value of \(T_i\) decreases by \(\dfrac{2}{5}\) for every increment of 1 in \(i\), forming a descending arithmetic sequence.

Utilizing the series formula lets you compute any term's position in the sequence and aids in efficiently calculating the desired partial sum using established properties of arithmetic series.