Question

Suppose \(\sum_{i=1}^{n} a_{i}=2, \sum_{j=1}^{m} b_{j}=3,\) and \(\sum_{k=1}^{l} c_{k}=5 .\) Evaluate $$ \sum_{i=1}^{n} a_{i}\left(\sum_{j=1}^{m} \sum_{k=1}^{l} b_{j} \cdot c_{k}\right) $$

Short answer

The answer is 30.

Step-by-step solution

Understand the Given Information

We have three sums: \( \sum_{i=1}^{n} a_{i} = 2 \), \( \sum_{j=1}^{m} b_{j} = 3 \), and \( \sum_{k=1}^{l} c_{k} = 5 \). We will use this information to evaluate the expression \( \sum_{i=1}^{n} a_{i}\left(\sum_{j=1}^{m}\sum_{k=1}^{l} b_{j} \cdot c_{k}\right) \).

Simplify the Inner Double Sum

Start by evaluating the double sum \( \sum_{j=1}^{m}\sum_{k=1}^{l} b_{j} \cdot c_{k} \). This is equivalent to the product \( \left(\sum_{j=1}^{m} b_{j}\right) \left(\sum_{k=1}^{l} c_{k}\right) = 3 \times 5 = 15 \).

Evaluate the Expression

Substitute the value from the previous step into the expression: \( \sum_{i=1}^{n} a_{i} \times 15 \). Since \( \sum_{i=1}^{n} a_{i} = 2 \), the expression becomes \( 2 \times 15 = 30 \).

Answer the Question

The final evaluated expression is \( 30 \). We have used the value from each summed sequence and combined it appropriately to reach this answer.

Key concepts

Summation

In discrete mathematics, summation refers to the operation of adding a sequence of numbers, typically expressed with the sigma notation \( \sum \). It is a way to represent the addition of a series of terms concisely. The subscript and superscript of the sigma sign define the range over which the terms are summed. For example, in \( \sum_{i=1}^{n} a_{i} \), the index \( i \) runs from 1 to \( n \), and the terms \( a_i \) are added together.
This concept is fundamental in various mathematical contexts, as it allows the compact expression of series and sequences. When summing, each term is calculated and added, one after another, making it crucial to follow the order dictated by the indices and limits.
Summations can be simple, with terms like consecutive integers, or complex, involving multiple nested layers. Nested or double summations, like \( \sum_{j=1}^{m}\sum_{k=1}^{l} b_{j} \cdot c_{k} \), involve evaluating sums inside other sums, which adds another layer of complexity. Understanding how to handle these expressions is vital for solving more complicated problems in discrete mathematics.

Mathematical Expression

Mathematical expressions are combinations of numbers, variables, operators, and functions that define certain values or results based on input. They are the building blocks of mathematical equations and often appear in coursework and problem-solving exercises.
The given expression \( \sum_{i=1}^{n} a_{i}\left(\sum_{j=1}^{m} \sum_{k=1}^{l} b_{j} \cdot c_{k}\right) \) combines multiple elements:

• The outer sum \( \sum_{i=1}^{n} a_{i} \)
• The nested sums \( \sum_{j=1}^{m} \sum_{k=1}^{l} b_{j} \cdot c_{k} \)

To solve such expressions, each component must be evaluated according to its operational hierarchy - starting from the innermost operations and moving outward.
Understanding how each piece interacts within the larger mathematical framework is essential. Plugging values into these expressions can simplify them significantly, reducing complex operations to straightforward arithmetic. This simplification often involves creating provisional results from smaller, nested expressions before incorporating them into the main expression.

Problem Solving

Problem solving in mathematics involves systematically approaching a question to find a solution. This process is integral to learning and applying discrete mathematics and requires a set of strategies and understanding of underlying concepts.
Here's a general approach based on the example given:

• **Understand** the problem: Break down what is given and what needs to be solved.
• **Plan** your steps: Identify actions to simplify complex portions, like nested sums.
• **Execute** the plan: Carry out each mathematical operation carefully to avoid errors.
• **Review** your work: Check each step to ensure correctness and reliability of the solution.

In this exercise, understanding started with recognizing the sum values provided. The planning involved simplifying a nested expression, leading to executing multiplication steps precisely. Finally, reviewing confirmed the result's accuracy – achieving the overall sum of 30.
By enhancing problem-solving skills, students can tackle diverse mathematical challenges effectively, developing a deeper understanding and better retention of mathematical concepts.