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Chapter 4: Problem 20

Compute the summations and products in 19-28 $$ \prod_{k=2}^{4} k^{2} $$

Short Answer

Expert verified
The given expression is a product notation, which requires us to compute the product of k squared for all k values between 2 and 4 (inclusive). Calculating the terms for each value of k in the specified range, we have 4, 9, and 16. The result of the given expression, \(\prod_{k=2}^{4} k^{2}\), is found by multiplying these terms together: \(4 \times 9 \times 16 = 576\).

Step by step solution

01

Understand the notation

The given expression is a product notation, which is represented as: \[ \prod_{k=2}^{4} k^{2} \] This means we need to compute the product of k squared for all k values between 2 and 4 (inclusive).
02

Calculate the terms

Now, we compute the terms for each value of k in the specified range: For k = 2: \[ (2)^{2} = 4 \] For k = 3: \[ (3)^{2} = 9 \] For k = 4: \[ (4)^{2} = 16 \]
03

Compute the product

Finally, we multiply the terms we calculated in the previous step: \[ \prod_{k=2}^{4} k^{2} = 4 \times 9 \times 16 = 576 \] So the result of the given expression is 576.

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Key Concepts

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

Product Notation
Product notation is a concise way to express the product of a sequence of numbers. It uses the \( \prod \) symbol, which looks like a large capital "Pi." This notation is particularly handy when we want to multiply several terms defined by a rule. For instance, in the notation \( \prod_{k=2}^{4} k^{2} \), the small numbers below and above the "Pi" indicate the range over which we're multiplying. The expression following these indices, \( k^{2} \), is the rule determining what is multiplied. In our example, it instructs us to multiply the squares of the numbers from 2 to 4.
Summation
Summation is a similar concept to product notation but involves addition instead. It is often represented by the \( \sum \) symbol, resembling a large "Sigma". Summations involve adding a series of numbers according to a specified rule. Despite not directly solving a summation in our original problem, understanding this concept helps distinguish between multiplying (products) and adding (summations). Both techniques can streamline how we work with sequences in mathematical problems. Like product notation, summation allows for a compact and clear representation of what could otherwise be cumbersome repetition.
Mathematical Notation
Mathematical notation helps us express complex ideas in a straightforward manner. Symbols like \( \prod \) and \( \sum \) serve as easily recognizable instructions for entire mathematical processes. They represent a universal language that allows mathematicians to read and solve problems consistently. The effectiveness of this notation lies in its precision and ability to convey detailed mathematical operations with just a few symbols. For students, learning this notation is critical because it provides the foundation for solving a wide range of mathematical problems, from simple arithmetic to more complex algebraic expressions.
Step-by-Step Solution
Approaching mathematical problems step-by-step is essential for clarity and accuracy. Let's go over our example:
  • First, identify the type of notation being used. Understanding whether it involves a product or summation is crucial.
  • Next, focus on calculating individual terms. For a product like \( \prod_{k=2}^{4} k^{2} \), compute \( k^{2} \) for each \( k = 2, 3, \) and \( 4 \).
  • Finally, multiply these computed terms together to arrive at the solution.
In our example, these steps guided us to find that the product of the squares from 2 to 4 is 576. Understanding each step ensures not only the correct answer but also deepens comprehension of the process.

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Most popular questions from this chapter

Write each of \(58-60\) as a single summation or product. $$ 2 \cdot \sum_{k=1}^{n}\left(3 k^{2}+4\right)+5 \cdot \sum_{k=1}^{n}\left(2 k^{2}-1\right) $$

For each positive integer \(n\), let \(P(n)\) be the property \(5^{n}-1\) is divisible by \(4 .\) a. Write \(P(0)\). Is \(P(0)\) true? b. Write \(P(k)\). c. Write \(P(k+0)\). d. In a proof by mathematical induction that this divisibility property holds for all integers \(n \geq 0\), what must be shown in the inductive step?

A sequence \(d_{1}, d_{2}, d_{3} \ldots\) is defined by letting \(d_{1}=2\) and \(d_{k}=\frac{d_{k-1}}{k}\) for all integers \(k \geq 2\). Show that for all integers \(n \geq 1, d_{n}=\frac{2}{n !}\)

Compute each of \(42-50 .\) $$ \frac{((n+1) !)^{2}}{(n !)^{2}} $$

\(1^{3}+2^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}\), for all integers \(n \geq 1\)
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