Problem 15 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

The sum is \( f(x_{1}) + f(x_{2}) + f(x_{3}) + f(x_{4}) + f(x_{5}) \).

Step 1: Understand the Summation

The expression \( \sum_{i=1}^{5} f(x_{i}) \) represents the sum of the function \( f(x) \) evaluated at each \( x_{i} \) from 1 to 5. Our task is to write this out explicitly without using Sigma (\( \Sigma \)) notation.

Step 2: Write Out Each Term

The summation notation \( \sum_{i=1}^{5} f(x_{i}) \) can be expanded into individual terms: \( f(x_{1}) + f(x_{2}) + f(x_{3}) + f(x_{4}) + f(x_{5}) \). This represents the sum of the function \( f \) evaluated at each \( x_{i} \) for \( i \) from 1 to 5.

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of variables, coefficients, and constants combined using operations like addition, subtraction, multiplication, and division. In the context of summation notation, an algebraic expression often includes a variable, like \( x_i \), which changes its value in a specific range. These expressions allow us to concisely represent mathematical relationships and calculations.
Understanding how to manipulate algebraic expressions is crucial, as it enables us to break down complex equations and systems into simpler parts. This is particularly helpful when transforming summation notation into explicit terms, which often involves expanding the algebraic expression for clarity. By practicing these transformations, students enhance their ability to work with and interpret more intricate algebraic problems.

Function Evaluation

Function evaluation is the process of determining the output of a function for specific inputs. It's a key concept when working with summation notation, as each term of the summation involves evaluating the function at different points.
To evaluate a function like \( f(x) \), substitute the input value (in this case, each \( x_i \)) into the function in place of the variable \( x \). For instance, if \( f(x) = 2x + 3 \) and you want to evaluate it at \( x_1 \), you replace \( x \) with \( x_1 \) to get \( 2x_1 + 3 \). Repeat this for each \( x_i \) to obtain individual terms for the summation.
Understanding function evaluation not only aids in transforming summation notation but also helps when solving real-world problems where evaluating a function at particular points is necessary.

Mathematical Notation

Mathematical notation is the symbolic language of mathematics. It includes a diverse set of symbols and rules for writing mathematical concepts and operations. Summation notation, involving the sigma (\( \Sigma \)) symbol, is a perfect example of how mathematical notation can simplify expressing complex operations.
Sigma notation specifically indicates the sum of a sequence of terms and is defined by specifying the start and end indices, along with the general term being summed. For example, \( \sum_{i=1}^{5} f(x_i) \) tells us to sum the function values from \( i=1 \) to \( i=5 \). The repeated use of symbols and a structured layout makes it easier to convey intricate mathematical ideas compactly and precisely.
Learning to interpret and use mathematical notation is a vital skill, enabling students to follow complex arguments, as well as communicate their own mathematical solutions effectively.