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Chapter 8: Problem 42

Find each indicated sum. $$\sum_{-1}^{5} \frac{(i+2) !}{i !}$$

Short Answer

Expert verified
It's impossible to find a factorial of a negative number, (-1 in this exercise's case), so at least one of the terms will be undefined and the sum can't be computed.

Step by step solution

01

Understand the Sequence

The sequence for this problem is given by \( \frac{(i+2) !}{i !} \), and we are asked to find the sum of all these terms for all 'i' from -1 to 5.
02

Calculate the terms of the Sequence

Start calculating each term. Plugging 'i' from -1 to 5, we will calculate these individual terms in the sequence: \( \frac{(1) !}{(-1) !} \), \( \frac{(2) !}{(0) !} \), \( \frac{(3) !}{(1) !} \), \( \frac{(4) !}{(2) !} \), \( \frac{(5) !}{(3) !} \), \( \frac{(6) !}{(4) !} \), \( \frac{(7) !}{(5) !} \).
03

Sum all the calculated terms

Now that we have calculated the values for all of these terms, the final step is to sum all these terms to get the answer.

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Key Concepts

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

Understanding Factorials
Factorials are a fundamental concept in mathematics, often represented by the symbol '!'.
A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \).
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are used in permutations, combinations, and various other mathematical calculations.
Special cases include \( 0! = 1 \) by definition. In this exercise, we see factorials used in the expression \( \frac{(i+2)!}{i!} \), which simplifies to \((i+2) \times (i+1)\).
  • \( (i+2)! \) is the factorial of \((i+2)\)
  • \( i! \) is divided out, simplifying calculations

By understanding these basics, we can easily compute each term in the given sequence.
Defining the Sequence
A sequence is an ordered list of numbers arranged according to a specific rule or pattern.
In this problem, we're dealing with a sequence defined by the expression \( \frac{(i+2)!}{i!} \).
This sequence changes with each different value of 'i'. For example, when \( i = -1 \), our term becomes \( \frac{(1)!}{(-1)!} \).
Sequences can be finite or infinite. Here it's finite, starting at \( i = -1 \) and ending at \( i = 5 \).
Each term depends on the previous calculation, thanks to the factorial function.
Understanding the rule behind a sequence helps us predict further terms and find sums accurately.
Calculating the Sum of Terms
The sum of terms in a sequence refers to the addition of all elements within the specified range.
In our exercise, we calculate terms from \( i = -1 \) to \( i = 5 \) using the expression \( \frac{(i+2)!}{i!} \) and then sum them.
  • First, compute each individual term: \( \frac{1!}{(-1)!}, \frac{2!}{0!}, \frac{3!}{1!}, \dots \)
  • This results in the terms being: \( 1, 2, 6, 12, 20, 30, 42 \)

Finally, all these values are added together to find the total sum. This method requires careful calculation, but once organized, it simplifies complex expressions.
Exploring the Mathematical Sequence
Mathematical sequences are everywhere in math, forming the foundation for functions and series.
They are not just numbers but follow a specific structure allowing prediction of future terms.
The sequence given is mathematical because each term can be clearly defined through the factorial relationship \( \frac{(i+2)!}{i!} \).
Mathematical sequences can describe real-world phenomena, like growth rates or financial forecasts.
Our specific sequence changes progressively, making it a valuable example of factorial and sequence interaction.
  • Allows understanding of patterns.
  • Simplifies complex problems into manageable parts.

This interplay is a key concept in higher-level mathematics and computer science.

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Most popular questions from this chapter

Write the first five terms of the sequence whose first term is 9 and whose general term is a. \(-\left\\{\begin{array}{ll}\frac{a_{-1}-1}{2} & \text { if } a_{-1} \text { is even } \\ 3 a_{-1}+5 & \text { i is } a_{-1} \text { is odd }\end{array}\right.\) for \(n \geq 2\)

Use the formula for \(_{n} C_{r}\) to solve Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January 10 , how many degree-days are included from January 1 to January \(10 ?\)

The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user.

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-5 \text { : Show that } 1+2+3+4+5-\frac{5(5+1)}{2}.$$
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