/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Find each indicated sum. $$\su... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 34

Find each indicated sum. $$\sum_{k=1}^{4}(k-3)(k+2)$$

Short Answer

Expert verified
The sum of the series is -4

Step by step solution

01

Understanding Sigma Notation

Sigma (危) notation is a way of writing down long sums in a concise way. In this problem, the series is described by the 危 notation \(\sum_{k=1}^{4}(k-3)(k+2)\). This means we are summing up the product \( (k-3)(k+2) \), for each value of k from 1 to 4.
02

Determine Each Term of the Series

We substitute k into \((k-3)(k+2)\) for each value from 1 to 4.\n For k=1: \((1-3)(1+2) = -2 \times 3 = -6\)\n For k=2: \((2-3)(2+2) = -1 \times 4 = -4\)\n For k=3: \((3-3)(3+2) = 0 \times 5 = 0\)\n For k=4: \((4-3)(4+2) = 1 \times 6 = 6\)
03

Sum up the Terms of the Series

Now, we add up the four terms we have calculated: -6, -4, 0, and 6. Adding these gives us \(-6 + (-4) + 0 + 6 = -4\)

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Key Concepts

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

Understanding Summation
Summation is a powerful mathematical concept used to add up a sequence of numbers. It is often represented using the Greek letter sigma (危), which is why it鈥檚 frequently called "sigma notation."
This notation is useful when dealing with patterns or sequences where each term needs to be added together. For example,
  • The expression \( \sum_{k=1}^{4}(k-3)(k+2) \) represents a summation where each term is obtained by substituting values of \( k \) from 1 to 4 into the expression \((k-3)(k+2) \).
  • It directs us to calculate the result step by step for each integer value of \( k \) and then sum these results.
Summation helps simplify complex calculations and is widely used in calculus and number theory.
Exploring Series
A series is the result of adding the terms of a sequence. When numbers follow a particular rule or pattern, their sum is called a series.
For the sequence given by \( (k-3)(k+2) \), we generated the series:
  • \( -6 \) for \( k=1 \)
  • \( -4 \) for \( k=2 \)
  • \( 0 \) for \( k=3 \)
  • \( 6 \) for \( k=4 \)
The sum of these numbers, \( -6 + (-4) + 0 + 6 \), results in the final total, which is known as the sum of the series and equals \(-4\).
Understanding series is crucial for solving problems involving cumulative sums and sequences in math.
Decoding Mathematical Notation
Mathematical notation is a symbolic language that allows us to express mathematical ideas concisely and clearly. It uses symbols, numbers, and letters to describe mathematical concepts such as equations, sums, and more.

In our problem, sigma notation \( \sum_{k=1}^{4}(k-3)(k+2) \) provides a structured way to solve the series. Here鈥檚 how:
  • The symbol \( \sum \) indicates summation.
  • \( k=1 \) to \( 4 \) is the range of values for the variable.
  • The expression \((k-3)(k+2)\) represents each term in the series.
By understanding these symbols and their functions, one can efficiently tackle complex mathematical problems without lengthy explanation.

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Most popular questions from this chapter

Some statements are false for the first few positive integers, but true for some positive integer \(m\) on. In these instances, you can prove \(S_{n}\) for \(n \geq m\) by showing that \(S_{m}\) is true and that \(S_{k}\) implies \(S_{k+1}\) when \(k \geq m .\) Use this extended principle of mathematical induction to prove that each statement in is true. Prove that \(n^{2} > 2 n+1\) for \(n \geq 3 .\) Show that the formula is true for \(n=3\) and then use step 2 of mathematical induction.

Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). $$\begin{array}{l} (a+b)^{1}=a+b \\ (a+b)^{2}=a^{2}+2 a b+b^{2} \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ (a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5} \end{array}$$ Describe the pattern for the exponents on \(a\).

Graph: \(f(x)=\frac{3 x-1}{x-1}\) (Section \(2.6,\) Example 5 )

In Exercises \(49-52,\) a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

In Exercises \(39-44\), you are dealt one card from a 52 -card deck. Find the probability that you are dealt a red 7 or a black 8 .
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