During a baseball season, a company pledges a donation to a charity of $5000 plus $100 for every home run hit by the local team. Does it make more sense to represent this situation using a sequence or a series? Explain your reasoning.
To determine whether it makes more sense to represent this situation using a sequence or a series, let's first understand the difference between the two.
A sequence is a list of numbers in a specific order, often denoted as (a₁, a₂, a₃, ...), where each number is related to the previous one by a specific rule or pattern.
A series is the sum of the terms of a sequence, often denoted as Σ (capital sigma symbol) followed by the sequence terms in brackets. It represents the cumulative sum of a given sequence.
In this scenario, we need to calculate the total amount of the donation based on the number of home runs hit by the local team.
Since the donation amount is determined by both a fixed pledge ($5000) and an additional amount ($100) for each home run, the number of home runs serves as the varying factor. Hence, we can represent this situation using a sequence.
We can build a sequence where each term represents the amount donated for each home run hit. For example, if the local team hits 5 home runs, the sequence would be (5000, 100, 100, 100, 100, 100).
To find the total donation amount, we can use the sequence to create a series and calculate the sum of all the terms. For the above example, the series would be Σ(5000, 100, 100, 100, 100, 100). By adding each term of the sequence together, the total donation can be calculated.
Therefore, while the situation can be represented as both a sequence (to indicate the individual amounts) and a series (to calculate the cumulative sum), it makes more sense to use a sequence as the primary representation since we want to showcase each individual donation amount for every home run hit.