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Chapter 4: Problem 21

Express \(\sum_{i=1}^{5} f\left(x_{i}\right)\) without using summation notation.

Short Answer

Expert verified
\(f(x_{1}) + f(x_{2}) + f(x_{3}) + f(x_{4}) + f(x_{5})\onumber \)

Step by step solution

01

Understand the summation notation

The given expression \(onumber \sum_{i=1}^{5} f(x_{i})\onumber \) represents the sum of the function values \(f(x_{i})\onumber \) as \(i\onumber \) ranges from 1 to 5.
02

List all terms in the summation

The terms involved in the summation are \(f(x_{1}), f(x_{2}), f(x_{3}), f(x_{4}), \text{\text{and}}\ f(x_{5})\onumber \).
03

Write out the summation as an explicit sum

Express the summation without summation notation by listing all the terms: \(f(x_{1}) + f(x_{2}) + f(x_{3}) + f(x_{4}) + f(x_{5})\onumber \).

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Key Concepts

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

explicit sum
In mathematics, the term 'explicit sum' refers to writing out every individual term of a summation. For example, the summation notation \( \sum_{i=1}^{5} f(x_{i}) \) can be expanded as follows: \( f(x_{1}) + f(x_{2}) + f(x_{3}) + f(x_{4}) + f(x_{5}) \). Breaking it down into an explicit sum helps us see and understand each part of the equation more clearly. This way, rather than dealing with a single compact notation, you unfold the entire series, making it visually simpler to grasp.
list terms
Listing terms means identifying each individual component of a summation. In our case, the summation \( \sum_{i=1}^{5} f(x_{i}) \) includes the terms \( f(x_{1}), f(x_{2}), f(x_{3}), f(x_{4}), \text\text{and}\ f(x_{5}) \). By listing them, we break down the summation step-by-step, allowing us to see what 'pieces' make up the whole. This approach is particularly useful in long or complex summations where each term contains valuable information.
mathematical notation
Understanding mathematical notation is key to decoding various expressions and formulas. Summation notation, represented by the symbol \( \sum \), is a way to compactly write the sum of a series of terms. In \( \sum_{i=1}^{5} f(x_{i}) \), the 'i' is called the index of summation, starting at 1 and ending at 5. The general form of summation notation is \( \sum_{i=a}^{b} f(x) \), which means adding up the function \( f(x) \) as 'i' runs from 'a' to 'b'. Simple notational tools like these help express complex ideas in concise and manageable forms. They are essential for communicating mathematical concepts efficiently.

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