Question

Write an expression in sigma notation for each situation. The total expenditure for a week if the daily expenditures are \(e_{1}, e_{2}, \ldots, e_{6}, e_{7}\)

Short answer

Question: Write the expression in sigma notation for the total expenditure of a week given the daily expenditures as \(e_1, e_2, \ldots, e_6, e_7\). Answer: \(\sum_{i=1}^{7} e_i\)

Step-by-step solution

Identify the indices of the daily expenditures

First, let's write down the indices for the daily expenditures: \(i = 1, 2, 3, \ldots, 6, 7\). These indices will be used in the sigma notation.

Write the sigma notation for the sum of daily expenditures

Now, we need to write the sum of the daily expenditures using the sigma notation. The sum can be written as: Total Expenditure = \(e_1 + e_2 + e_3 + \cdots + e_6 + e_7\) We can represent this sum using the sigma notation as follow: Total Expenditure = \(\sum_{i=1}^{7} e_i\) This is the expression in sigma notation for the total expenditure for a week, where \(e_i\) represents the expenditure for day \(i\).

Key concepts

Summation

Summation is a mathematical concept used to express the addition of a sequence of numbers. In this context, it allows us to combine multiple terms into a compact form. Whether we are adding numbers manually or using tools like sigma notation, we are performing a summation.

• The basic idea is to add all the numbers in a set together.
• This can involve a list of numbers, variables, or expressions.

Using summation can greatly simplify problems where multiple terms of the same nature are being added. This is essential in many mathematical fields including algebra, calculus, and statistics. By writing sums just in this folded way, we can handle large amounts of data more effectively and communicate complex ideas clearly.

Indices

Indices are numbers that indicate the position or location of terms within a sequence. In our problem, the daily expenditures are listed using indices. When constructing formulas or equations, indices help you specify each term clearly by its position. For example:

• If we have indices as 1, 2, 3, ..., 7, each corresponds to a specific day of the week.
• The index 'i' refers to the individual day expenditure, such as \(e_1\) for day 1, \(e_2\) for day 2, and so on.

Using indices allows us to work efficiently with lists and sums because we don't have to specify each term individually. Indices enable us to harness the power of symbolic notation, improving clarity and reducing the risk of errors in mathematical operations.

Sigma Notation Expression

Sigma notation is a concise way of representing a series or a sum of multiple terms. The sigma symbol \(\Sigma\) is used to denote the sum of a sequence, and it includes a lower and upper limit indicating where the summation starts and ends.

• The expression \(\sum_{i=1}^{7} e_i\) exemplifies sigma notation, where \(\Sigma\) tells us 'to sum up'.
• This notation reads as "the sum of \(e_i\) from \(i = 1\) to \(i = 7\)".
• The lower limit of the index (1) denotes the starting point, while the upper limit (7) represents the endpoint of the summation.

Sigma notation effectively replaces elongated expressions into abbreviated and manageable formulas, making complicated mathematical expressions easier to interpret and utilize. It's particularly useful in programming, algebra, and statistics, facilitating the calculation of sums over sequences without repetitive manual addition.

Algebraic Representation

Algebraic representation refers to expressing mathematical ideas and relationships with symbolic notation. In our situation, we use symbols to write the sum of expenditures in a week. Each expenditure is depicted by \(e_i\), where \('i'\) ranges over the days of the week. This allows us to efficiently convey the pattern repeated over several elements.

• In our example, \(e_i\) signifies the expenditure on day \(i\).
• The use of algebraic expressions helps in generalizing specific instances of expenditure into a single, powerful notation.
• By writing \(\sum_{i=1}^{7} e_i\), we summarize all daily expenditures clearly and concisely.

Such representations not only simplify problem-solving but also illustration of mathematical relationships. Algebraic representation provides a framework that translates practical problems into mathematical language, which forms the foundation of solving equations, generating functions, and modeling scenarios.