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Chapter 11: Problem 74

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{6}\left(2+i-i^{2}\right)$$

Short Answer

Expert verified
-58

Step by step solution

01

- Expand and Separate the Summation

Use the linearity property of summations to separate the terms inside the summation oindent\[\sum_{i=1}^{6} \left( 2 + i - i^2 \right) = \sum_{i=1}^{6} 2 + \sum_{i=1}^{6} i - \sum_{i=1}^{6} i^2\]
02

- Evaluate the Summation for 2

The summation of 2 from 1 to 6 is oindent\[\sum_{i=1}^{6} 2 = 2 \times 6 = 12\]
03

- Evaluate the Summation for i

The sum of the first n natural numbers is given by the formula: \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\). So for this problem:oindent\[\sum_{i=1}^{6} i = \frac{6 \times (6 + 1)}{2} = \frac{6 \times 7}{2} = 21\]
04

- Evaluate the Summation for i^2

The sum of the squares of the first n natural numbers is given by the formula: \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\). So for this problem:oindent\[\sum_{i=1}^{6} i^2 = \frac{6 \times 7 \times 13}{6} = 91\]
05

- Combine the Results

Combine the results from the previous steps:oindent\[\sum_{i=1}^{6} \left( 2 + i - i^2 \right) = 12 + 21 - 91 = -58\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

series evaluation
Series evaluation is the process of calculating the sum of a sequence of numbers, often represented by a summation. In this exercise, we evaluate the series \(\sum_{i=1}^{6} (2 + i - i^2)\) using summation properties. The linearity of summation allows us to separate the series into simpler parts: \[\sum_{i=1}^{6} (2 + i - i^2) = \sum_{i=1}^{6} 2 + \sum_{i=1}^{6} i - \sum_{i=1}^{6} i^2 \]. Breaking down the series:
  • The first part is the summation of a constant, 2.
  • The second part is the summation of natural numbers, represented by i.
  • The third part involves the summation of squared terms, i虏.
These can be simplified and evaluated individually, then combined to get the final result. Each step relies on known summation formulas to ensure accurate and efficient calculation.
natural numbers
Natural numbers are positive integers starting from 1 and moving upwards (1, 2, 3,...). They form the basis of many mathematical operations and are crucial in evaluating series. In this problem, the summation of natural numbers \(\sum_{i=1}^{n} i\) is calculated using the formula \[\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\]. Applying this to our series where n = 6: \[\sum_{i=1}^{6} i = \frac{6 \times (6 + 1)}{2} = 21\]. Understanding this formula is essential because it simplifies the process of adding up natural numbers within any range, ensuring you can quickly find the sum without manual counting.
sum of squares
The sum of squares concept deals with summing the squares of natural numbers up to a specific limit. The summation of squares formula is given as \[\sum_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}\]. This is handy when evaluating series with squared terms. For our problem: \[\sum_{i=1}^{6} i^2 = \frac{6 \times 7 \times 13}{6} = 91\]. By using this formula, you can efficiently find the sum of squares for any number range, saving time and reducing errors. Summing squared terms is common in various mathematical contexts, making this a valuable tool in your problem-solving toolkit.

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Most popular questions from this chapter

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Four financial planners are to be selected from a group of 12 to participate in a special program. In how many ways can this be done? In how many ways can the group that will not participate be selected?

The table gives the results of a 2008 survey of Americans aged \(18-24\) in which the respondents were asked, "During the past 30 days, for about how many days have you felt that you did not get enough sleep?"$$\begin{array}{|l|c|c|c|c|}\hline \text { Number of Days } & 0 & 1-13 & 14-29 & 30 \\ \hline \text { Percent (as a decimal) } & 0.23 & 0.45 & 0.20 & 0.12 \\\\\hline\end{array}$$ Using the percents as probabilities, find the probability that, out of 10 respondents in the \(18-24\) age group selected at random, the following were true. No more than 3 did not get enough sleep on \(1-29\) days.

According to T. Rowe Price Associates, a person with a moderate investment strategy and \(n\) years to retirement should have accumulated savings of \(a_{n}\) percent of his or her annual salary. The geometric sequence $$a_{n}=1276(0.916)^{n}$$ gives the appropriate percent for each year \(n\) (a) Find \(a_{1}\) and \(r .\) Round \(a_{1}\) to the nearest whole number. (b) Find and interpret the terms \(a_{10}\) and \(a_{20} .\) Round to the nearest whole number.

For students taking a course in black-and-white photography, the final step in processing a print is to immerse it in a chemical fixer. The print is then washed in running water. Under certain conditions, \(98 \%\) of the fixer in a print will be removed with 15 min of washing. How much of the original fixer would be left after 1 hr of washing?
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