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Chapter 5: Problem 17

Suppose that \(\sum_{k=1}^{n} a_{k}=-5\) and \(\sum_{k=1}^{n} b_{k}=6 .\) Find the values of a. \(\sum_{k=1}^{n} 3 a_{k}\) b. \(\sum_{k=1}^{n} \frac{b_{k}}{6} \quad\) c. \(\sum_{k=1}^{n}\left(a_{k}+b_{k}\right)\) d. \(\sum_{k=1}^{n}\left(a_{k}-b_{k}\right) \quad\) e. \(\sum_{k=1}^{n}\left(b_{k}-2 a_{k}\right)\)

Short Answer

Expert verified
a. -15; b. 1; c. 1; d. -11; e. 16.

Step by step solution

01

Calculate \( \sum_{k=1}^{n} 3a_k \)

The given sum is \( \sum_{k=1}^{n} a_k = -5 \). To find \( \sum_{k=1}^{n} 3a_k \), we can factor out the constant from the sum: \( 3 \times \sum_{k=1}^{n} a_k = 3 \times (-5) = -15 \). Thus, \( \sum_{k=1}^{n} 3a_k = -15 \).
02

Calculate \( \sum_{k=1}^{n} \frac{b_k}{6} \)

The given sum is \( \sum_{k=1}^{n} b_k = 6 \). To find \( \sum_{k=1}^{n} \frac{b_k}{6} \), factor out the constant: \( \frac{1}{6} \times \sum_{k=1}^{n} b_k = \frac{1}{6} \times 6 = 1 \). Thus, \( \sum_{k=1}^{n} \frac{b_k}{6} = 1 \).
03

Calculate \( \sum_{k=1}^{n} (a_k + b_k) \)

The expression involves the sum of two sequences. Use the properties of summation: \( \sum_{k=1}^{n} (a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k = -5 + 6 = 1 \).
04

Calculate \( \sum_{k=1}^{n} (a_k - b_k) \)

Similarly, use the properties of summation: \( \sum_{k=1}^{n} (a_k - b_k) = \sum_{k=1}^{n} a_k - \sum_{k=1}^{n} b_k = -5 - 6 = -11 \).
05

Calculate \( \sum_{k=1}^{n} (b_k - 2a_k) \)

Here, apply the properties of summation: \( \sum_{k=1}^{n} (b_k - 2a_k) = \sum_{k=1}^{n} b_k - 2 \times \sum_{k=1}^{n} a_k = 6 - 2 \times (-5) = 6 + 10 = 16 \).

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Key Concepts

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

Series
A series is a sum of terms in a sequence. Imagine you have a sequence, which is a list of numbers that follows a certain rule. When we add these numbers together, we get a series. For example, if you have a sequence of 1, 2, 3, the series would be 1 + 2 + 3. It's all about turning the list of numbers into one big sum.
  • A sequence is like having notes for a song, while the series is playing those notes together in harmony.
  • The order of adding doesn't matter; what matters is getting the total sum.
This makes series a powerful tool for finding the total of numbers that follow a pattern, such as the cases we solve in math problems or even real-world situations like calculating total sales over several years.
Sequence
Before adding up terms in a series, we start with a sequence. A sequence is an ordered list of numbers and can follow various patterns or rules. For instance, a sequence could be as simple as an arithmetic progression like 2, 4, 6, 8, where each number increases by 2. Or, it could be more complex, like a geometric progression.
  • A sequence has a distinct order; changing this order can create a new sequence entirely.
  • Think of a sequence as a step-by-step list that must be followed in order.
Recognizing the pattern of a sequence allows us to predict subsequent terms and aids in solving problems that involve series, such as finding the sum of the sequence.
Summation Properties
Summation properties include rules that simplify calculating the sum of sequences, making it easier to handle complex calculations. These properties help us add multiple sequences together, or even subtract them, by splitting the sums around an addition or subtraction sign.
Some key summation properties include:
  • Distributive Property: This property allows us to factor out constants from a summation. For example, if we have 3 times the sum of sequence \(a_k\), it becomes 3 multiplied by the sum of the \(a_k\). This makes calculations straightforward.
  • Summation of Sums: If you are summing expressions like \( (a_k + b_k) \), you can sum each part separately and then combine them: \(\sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k \).
Understanding these properties is like having shortcuts, helping you solve problems quickly and efficiently.
Factorization of Constants
The factorization of constants in a summation equation is an important technique to simplify solving problems. It allows constants to be pulled out of a sum, which can be particularly useful in complex series. Let's say you have a constant multiplying each term in your series, like \(3a_k\). Instead of adding each \(3a_k\) individually, you can factor out the 3 and multiply the sum of \(a_k\) by 3.
  • This method significantly reduces the number of operations you need to perform.
  • It's like pulling out a common factor in an equation to simplify it.
Factorizing constants makes tough math problems more accessible, focusing only on necessary calculations while ignoring repetitive ones. It's a neat trick that鈥檚 not only efficient but also simplifies expressions greatly.

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Most popular questions from this chapter

If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals .Use \(n=4,10,20,\) and 50 subintervals of equal length in each case. $$\int_{0}^{\pi / 4} \sec ^{2} x d x=1$$

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and \(f .\) Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F\) b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrem and the points of inflection of \(F\) d. Using the information from parts (a)-(c), draw a rough hand sketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch.$$a=1, \quad u(x)=x^{2}, \quad f(x)=\sqrt{1-x^{2}}.$$

Find the total area between the region and the \(x\) -axis. $$y=-x^{2}-2 x,-3 \leq x \leq 2.$$

The region bounded below by the parabola \(y=x^{2}\) and above by the line \(y=4\) is to be partitioned into two subsections of equal area by cutting across it with the horizontal line \(y=c\) a. Sketch the region and draw a line \(y=c\) across it that looks about right. In terms of \(c,\) what are the coordinates of the points where the line and parabola intersect? Add them to your figure. b. Find \(c\) by integrating with respect to \(y\). (This puts \(c\) in the limits of integration.) c. Find \(c\) by integrating with respect to \(x\). (This puts \(c\) into the integrand as well.)

Find the area of the region in the first quadrant bounded on the left by the \(y\) -axis, below by the line \(y=x / 4,\) above left by the curve \(y=1+\sqrt{x}\), and above right by the curve \(y=2 / \sqrt{x}\).
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