Sequences and series, whether they be arithmetic or geometric, have may applications to situations you may not think of as being related to sequences or series. In order to work with these application problems you need to make sure you have a basic understanding of arithmetic sequences, arithmetic series, geometric sequences, and geometric series. Let's Practice: An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. How many seats are in the theatre?
To solve this problem, we need to ask and answer some preliminary questions.
First, what is the problem asking us to do? We need to know how many seats are in the auditorium which means we are counting things and finding a total. In other words, we need to add up all the seats on each row. Since we are adding things up, this can be looked at as a series. We have formulas for series problems, but we have to know if it is arithmetic or geometric so we’ll know which formula to use.
To know if it’s arithmetic or geometric, look at the pattern in the problem. There are 20 seats on the first row, 24 on the second row, and 28 on the third row. Each row has four more seats than the one before it. Since we are adding four to each row, this is an arithmetic sequence of numbers that we will be adding up.
So we now know that our goal is to find an arithmetic series. The formula for an arithmetic series is
To solve this problem we need n, a_{1}, and a_{n}. In this problem, n will be equal to 30 because we are being asked to find how many seats are in all 30 rows. Or to add up the seats in the 30 rows. The first term in the sequence, a_{1}, is 20 because the problem tells us that the first row has 20 seats. The only thing left to do is find a_{n }which will be a_{30}.
To find a_{30 }we need the formula for the sequence and then substitute n = 30. The formula for an arithmetic sequence is We already know that is a_{1} = 20, n = 30, and the common difference, d, is 4. So now we have So we now know that there are 136 seats on the 30^{th} row. We can use this back in our formula for the arithmetic series.  Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second day, 4 cents on the third day and so on. If the daily wage keeps doubling, what will you total income be for working 31 days?
This problem is geometric because the problem says that the salary from the previous day is doubled, or multiplied by 2. When we are multiplying by the same number each time, this is a geometric sequence. But what do we need to do with this geometric sequence?
The problem wants to know TOTAL income after 31 days. When dealing with total amounts, like in the previous example, we need to add the terms in a sequence. In this case, since we will be adding terms in a geometric sequence, we will be finding a geometric series. So we need the formula for a geometric series. We need to know n, a_{1}, and r. We are told r = 2 when the problem says doubling and n = 31 since that’s how many things we need to add up. We also know the first term which is 0.01 (the decimal amount for one penny). This should give us enough information to find the answer. Almost 21 1/2 million dollars total for 31 days work. Not a bad job! 

