Question

Write an expression in sigma notation for each situation. The total rainfall for the month of September if the rainfall on the \(i^{\text {th }}\) day is \(r_{i}\)

Short answer

Question: Write an expression in sigma notation to represent the total rainfall in the month of September. Answer: $$\sum_{i=1}^{30} r_i$$

Step-by-step solution

Identify the variable and index

We are given that the total rainfall on the \(i^{\text{th}}\) day is \(r_i\), where \(i\) shows the sequence of days from the first to the thirtieth day.

Determine the limits of the index

The month of September has 30 days, so we have to sum the rainfalls from the first day \((i=1)\) to the thirtieth day \((i=30)\).

Writing the summation notation

We sum up the rainfalls on the \(i^{\text{th}}\) day from day 1 to day 30 using the sigma notation. The expression for the total rainfall in the month of September is: $$\sum_{i=1}^{30} r_i$$

Key concepts

Summation

In mathematics, summation refers to the operation of adding a sequence of numbers. This sequence can be finite or infinite. When dealing with summation, we often want to calculate the total sum of numerous values quickly. Instead of writing out the full addition of all individual terms, we use an efficient notation called sigma notation. Sigma notation is represented by the Greek letter 'Σ', and it is a shorthand for expressing a sum of terms following a pattern. This can simplify calculations significantly, especially when dealing with large datasets or series of numbers. In our exercise, summation helps us efficiently calculate the total rainfall by adding up the daily rainfall amounts over the month of September.

Index Variable

The index variable in a summation is a placeholder that represents each term in the series being summed. Often denoted by letters such as 'i', 'j', or 'k', the index variable takes on successive integer values as specified by the limits of summation. In our exercise, the index variable is denoted by 'i'. It represents the day of the month for which we are calculating daily rainfall, thereby running from 1 (for September 1st) to 30 (for September 30th). The index variable is crucial because it allows us to systematically refer to each term in the series, in this case, each day's rainfall. By changing its value, the index variable helps iterate through all the units that need to be added together.

Limits of Summation

Limits of summation define the starting and ending points for the index variable over which summation is performed. These are crucial as they specify the scope of the terms you want to add.In the given problem, the limits of summation are provided for the 30 days of September. The lower limit corresponds to the first day of the month \(i=1\) and the upper limit to the thirtieth day \(i=30\). The sigma notation takes these limits and systematically iterates from the starting index to the ending index, calculating the sum of all values of the expression for each day covered by these bounds.

Expression

In the context of sigma notation, an expression is the mathematical phrase applied to each value of the index variable as it iterates through the specified limits. This expression determines what is actually being summed.For our rainfall problem, the expression is denoted as \( r_i \). Here, \( r_i \) represents the rainfall amount on the \( i^{\text{th}} \) day of September. Therefore, as the index 'i' runs from 1 to 30, \( r_i \) refers to each day's rainfall amount, and this forms the specific values being added together in the summation. The clarity of the expression is important because it tells us explicitly what is being counted in the summation calculation.