Question

In a survey of \(n\) gas stations, the price of gas at the \(i^{\text {th }}\) gas station is \(p_{i}\) dollars/gallon. Write an expression for (a) The cost of buying 12 gallons of gas at the \(i^{\text {th }}\) gas station. (b) The price of gas at the second to last gas station surveyed. (c) The average price at all \(n\) gas stations.

Short answer

Answer: The cost of buying 12 gallons of gas at the \(i^{\text{th}}\) gas station is \(12 \cdot p_i\), the price of gas at the second to last gas station is \(p_{n-1}\), and the average price at all \(n\) gas stations is \(\dfrac{\sum_{i=1}^{n} p_i}{n}\).

Step-by-step solution

(a) Cost of buying 12 gallons of gas at the i-th station

To find the cost of buying 12 gallons of gas at the \(i^{\text{th}}\) gas station, we simply need to multiply the price per gallon at that station, \(p_i\), by the number of gallons bought. This can be represented as: \(12 \cdot p_i\)

(b) Price of gas at the second to last gas station

To find the price of gas at the second to last gas station surveyed, we first need to determine the index of that gas station in the series of \(n\) gas stations. The second to last gas station has an index \(n-1\), because the last gas station has an index of \(n\). Therefore, the price at the second to last gas station is represented by: \(p_{n-1}\)

(c) Average price at all n gas stations

To find the average of the prices at all \(n\) gas stations, we need to sum the prices of all gas stations and then divide by the total number of gas stations. We can represent the sum of the prices as the summation \(\sum_{i=1}^{n} p_i\), and the total number of gas stations is \(n\). Therefore, the average price at all \(n\) gas stations can be represented as: \(\dfrac{\sum_{i=1}^{n} p_i}{n}\)

Key concepts

Summation Notation

Summation notation is a way to represent the sum of a series of numbers. It is often used in algebra to simplify expressions where you need to add up multiple terms. The Greek letter sigma (\( \Sigma \)) is used to denote this summation process.

• The expression \( \sum_{i=1}^{n} p_i \) signifies summing the prices \( p_i \) for gas at each of the \( n \) gas stations.
• The lower limit \( i=1 \) tells us where to start the summation, and \( n \) is where to stop.

This notation is helpful because it condenses a potentially long series of arithmetic operations into a single expression. Instead of writing out each addition, you use the summation notation to express it compactly. This concept is crucial when calculating averages or dealing with sequences in math and statistics.

Indices in Sequences

Indices in sequences refer to the position of an element within a series, like a list of numbers or a series of prices. These indices are essential for identifying specific elements and performing operations on them. In our exercise example:

• Each gas station has an index denoted as \( i \).
• The second to last station, before the last one, has the index \( n-1 \).
• Understanding indices helps in identifying terms accurately, like determining \( p_{n-1} \) for the gas price at the second last station.

In sequences, mastering how to use indices can simplify complex problems, as it allows you to point directly to the needed data without ambiguity. Moreover, indices can be dynamic, adjusting as sequences are modified or as more elements are added.

Average Calculation

The average of a set of numbers is a fundamental concept in mathematics. It gives us an idea of the central or typical value of a dataset. To find the average price across all \( n \) gas stations:

• First, sum up all the prices using summation notation: \( \sum_{i=1}^{n} p_i \).
• Then, divide the total by \( n \), the number of gas stations: \( \frac{\sum_{i=1}^{n} p_i}{n} \).

This operation gives us the mean price of gas at these stations. Calculating the average is vital for assessments, like comparing current data with historical trends or benchmarking against other locations. It simplifies a large dataset into a single, more manageable figure, providing essential insight in an accessible form.