Question

Expand the following summation expressions \((a) \sum_{i=2}^{5} x_{i}\) (b) \(\sum_{i=5}^{8} a_{i} x_{i}\) \((c) \sum_{i=1}^{4} b x_{i}\) \((d) \sum_{i=1}^{n} a_{i} x^{i-1}\) \((e) \sum_{i=0}^{3}(x+i)^{2}\)

Short answer

(a) \( x_{2} + x_{3} + x_{4} + x_{5} \); (b) \( a_{5}x_{5} + a_{6}x_{6} + a_{7}x_{7} + a_{8}x_{8} \); (c) \( b x_{1} + b x_{2} + b x_{3} + b x_{4} \); (d) \( a_{1} x^{0} + a_{2} x^{1} + \ldots + a_{n} x^{n-1} \); (e) \( x^{2} + (x+1)^{2} + (x+2)^{2} + (x+3)^{2} \).

Step-by-step solution

Expand the first summation expression (a)

Consider the expression \( \sum_{i=2}^{5} x_{i} \). This expression means summing individual terms \( x_{i} \) as \( i \) takes values from 2 to 5. Therefore, this expands to: \( x_{2} + x_{3} + x_{4} + x_{5} \).

Expand the second summation expression (b)

The expression \( \sum_{i=5}^{8} a_{i} x_{i} \) denotes summing the terms \( a_{i} x_{i} \) from \( i = 5 \) to \( i = 8 \). Thus, the expanded expression is: \( a_{5}x_{5} + a_{6}x_{6} + a_{7}x_{7} + a_{8}x_{8} \).

Expand the third summation expression (c)

We analyze the expression \( \sum_{i=1}^{4} b x_{i} \). Here, \( b \) is constant, so it multiplies each \( x_{i} \) term as \( i \) varies from 1 to 4. Therefore, the expanded sum is: \( b x_{1} + b x_{2} + b x_{3} + b x_{4} \).

Expand the fourth summation expression (d)

The expression \( \sum_{i=1}^{n} a_{i} x^{i-1} \) involves summing products of each \( a_{i} \) and \( x^{i-1} \), with \( i \) spanning from 1 to \( n \). The expansion is \( a_{1} x^{0} + a_{2} x^{1} + a_{3} x^{2} + \, \ldots \, + a_{n} x^{n-1} \).

Expand the fifth summation expression (e)

For \( \sum_{i=0}^{3}(x+i)^{2} \), calculate and sum the squares of \( x+i \) for \( i = 0 \) to 3. Thus, this expands to: \( (x+0)^{2} + (x+1)^{2} + (x+2)^{2} + (x+3)^{2} \). Evaluating these, you get \( x^{2} + (x+1)^{2} + (x+2)^{2} + (x+3)^{2} \).

Key concepts

Summation expressions

Summation expressions are mathematical representations used to sum a sequence of numbers, where the index of summation indicates the range and terms to sum over. They are typically written using the summation symbol \( \sum \) followed by an expression that indicates what terms should be included in the sum. In a summation like \( \sum_{i=2}^{5} x_{i} \), the index \( i \) starts from 2 and goes to 5, summarizing the elements \( x_2, x_3, x_4, \) and \( x_5 \).

Summation expressions are convenient in mathematical economics for compactly describing operations where many similar calculations need to be performed, such as calculating total cost, total utility, or summing products of price and quantity across different goods.

To read a summation expression:

• Find the lower bound (starting point) and the upper bound (ending point) of the index.
• Identify the term to be summed, which might be a variable or a more complex expression.
• Perform the addition of all the evaluated terms based on the index range.

Understanding these basics helps in grasping more advanced topics like series expansions and is an essential skill in mathematical economics.

Series expansion

Series expansion refers to expressing a mathematical function as a sum of terms derived from its elements. It's often used to approximate complex functions using simpler ones like polynomials. This is prevalent in mathematical economics for simplified model formulation and analysis.

A practical example using series expansion in economics might involve expanding a cost function. Suppose a series expansion transforms a total cost into simpler terms to see how small changes in production influence costs.
Consider the series \( a_{1} x^{0} + a_{2} x^{1} + a_{3} x^{2} + \ldots + a_{n} x^{n-1} \). Each term is calculated for a power of \( x \) starting from zero. These can model economic concepts where variables change dynamically.

Series expansions aid in breaking down more complex expressions into manageable parts. This process helps students and professionals understand the impact of different variables independently and in combination, which is crucial for economic analysis.

Mathematical notation

Mathematical notation is the language through which mathematical ideas are communicated. It uses symbols and formulas to describe numbers, operations, relationships, and structures clearly and succinctly.

Consider the summation \( \sum_{i=0}^{3}(x+i)^2 \). The notation \( \sum \) indicates a sum, the subscript and superscript define the start and end index, and the parentheses and operations describe what to calculate in each term.

• \( \sum \) implies summation.
• The subscript and superscript \( i=0 \) to 3 denote which elements are involved.
• \( (x+i)^2 \) defines how to modify \( x \) through each term's power of 2.

The efficiency of mathematical notation arises from its ability to convey complex calculations in a compact form.

It's crucial to understand these notations as they are widely used in textbooks and real-world applications, thus forming the backbone of analytical problem-solving in economics.

Summation techniques

Summation techniques are methods used to evaluate the sum of sequences, finite or infinite. These techniques are crucial in mathematics, especially in analyzing series that appear in economics to calculate total costs, revenue, or aggregate demand.

The primary techniques include:

• Direct Summation: Sum each term individually as shown in \( x_{2} + x_{3} + x_{4} + x_{5} \) where each term is evaluated individually.
• Arithmetic Series: Utilized when summands follow an arithmetic progression.
• Geometric Series: Applied when terms form a geometric progression. Useful in calculating present value in economics.
• Computational Tools: Complex summations might require computational tools or software for exact calculations.

Understanding these techniques provides a solid foundation for solving problems involving repeated addition of terms. Developing efficient summation methods can lead to breakthroughs in economic modeling and data analysis, ensuring accurate and efficient computations.