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Chapter 5: Problem 17

Express the sums in closed form. $$ \sum_{k=1}^{n} \frac{3 k}{n} $$

Short Answer

Expert verified
\( \frac{3(n+1)}{2} \)

Step by step solution

01

Distribution of Constant Multiplier

Factor out the constant multiplier from the summation. In this case, the constant is \( \frac{3}{n} \). Thus, the expression becomes: \[ \sum_{k=1}^{n} \frac{3k}{n} = \frac{3}{n} \sum_{k=1}^{n} k \]
02

Replace the Sum with Its Formula

Recognize that \( \sum_{k=1}^{n} k \) is the well-known formula for the sum of the first \( n \) positive integers, which is \( \frac{n(n+1)}{2} \). Substitute this formula in: \[ \frac{3}{n} \sum_{k=1}^{n} k = \frac{3}{n} \cdot \frac{n(n+1)}{2} \]
03

Simplify the Expression

Simplify the expression by carrying out the multiplication, \( \frac{3}{n} \cdot \frac{n(n+1)}{2} = \frac{3n(n+1)}{2n} \). Notice that the \( n \) in the numerator and the denominator cancel each other out, leaving: \[ \frac{3(n+1)}{2} \]
04

Conclusion

The closed form expression for the sum \( \sum_{k=1}^{n} \frac{3k}{n} \) is simply \( \frac{3(n+1)}{2} \).

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Key Concepts

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

Sum of Integers
Understanding the sum of integers plays a crucial role when learning about sequences and series in mathematics. When you see a sum like \( \sum_{k=1}^{n} k \), it refers to the sum of all integers starting from 1 up to \( n \). The concept might seem daunting at first, but remember this simple breakdown:
- You are adding numbers sequentially; think of 1 + 2 + 3 all the way up to your chosen \( n \).
- The sum formula \( \frac{n(n+1)}{2} \) provides an elegant way to calculate this total quickly, without manual addition.
  • "\( n \)" is the largest integer in your sequence.
  • The term "\( (n+1) \)" prepares for the sum total stretching one place further.
It's a beautiful example of how numbers have inherent patterns that we can describe effectively using algebra.
Summation Formulas
Summation formulas give us a toolset to work efficiently with sums in many mathematical contexts. In the problem presented, you encountered \( \sum_{k=1}^{n} \frac{3k}{n} \). Here are the steps that used the power of a summation formula:
- First, by recognizing constants, which can be factored out of the summation: the fraction \( \frac{3}{n} \) is a constant and can be moved outside of the summation.
- Then, applying the well-known formula for the sum of the first \( n \) integers, \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \).

These formulas help break down and simplify larger, more cumbersome expressions, providing a quicker path to solutions, avoiding tedious calculations.
Mathematical Simplification
Mathematical simplification is about reducing complex expressions to their most concise and understandable form. Once you've applied a summation formula, such as with our ongoing example, the next goal is to simplify:
- Begin with multiplication of simplified parts: notice where terms can cancel each other out, such as \( \frac{3}{n} \cdot \frac{n(n+1)}{2} = \frac{3n(n+1)}{2n} \).
- Since \( n \) appears in both the numerator and denominator, it can be cancelled, leaving the expression easier to interpret: \( \frac{3(n+1)}{2} \).
  • This emphasizes how simplification not only makes expressions easier to handle but also helps reveal any underlying meaning or outcome.
By simplifying, you often find an expression's closed form, which is a more elegant and insightful conclusion to a problem-solving process.

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Most popular questions from this chapter

$$ \int x^{2} e^{-2 x^{3}} d x $$

(a) Prove that if \(f\) is an odd function, then $$ \int_{-a}^{a} f(x) d x=0 $$ and give a geometric explanation of this result. [Hint: One way to prove that a quantity \(q\) is zero is to show that \(q=-q .]\) (b) Prove that if \(f\) is an even function, then $$ \int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x $$ and give a geometric explanation of this result. [Hint: Split the interval of integration from \(-a\) to \(a\) into two parts at \(0 .]\)

Evaluate each integral by first modifying the form of the integrand and then making an appropriate substitution, if needed $$ \int \cot x d x $$

Use a trigonometric identity to evaluate the integral. $$ \int \tan ^{2} x d x $$

An artist wants to create a rough triangular design using uniform square tiles glued edge to edge. She places \(n\) tiles in a row to form the base of the triangle and then makes each successive row two tiles shorter than the preceding row. Find a formula for the number of tiles used in the design. [Hint: Your answer will depend on whether \(n\) is even or odd.]
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