WEEK ONE

Week One Writing Assignment
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MAT 126
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WEEK ONE
Week One Writing Assignment In the 5th chapter of Mathematics in our world, we have read about sequence numbers. We went over two basic types of sequence numbers which are arithmetic and geometric sequence. Both sequence styles have their own rule that must be followed in order to determine what category the sequence falls in. By following these rules we can successfully solve real life problems. The first sequence we will discuss is the arithmetic sequence. Each single number in the arithmetic sequence is called a term. Every term is evenly spaced by the same amount i.e, 2, 4, 6, 8… are evenly spaced by adding 2 and 5, 10, 15, 20… are spaced by adding 5. This number that spaces or succeeds the term is known as the common difference. The common difference stays the same throughout the sequence. It is generally written as a1, a2, a3,…,an, where a1 is the first term. Here is an example of real world arithmetic sequence problem where we will find the term.
“A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of the labor for each succeeding 10 feet is $25 more that the preceding 10 feet. That is, the next 10 feet will cost $125; the next 10 feet will cost $150, etc. How much will it cost to build a 90-foot tower?” (Bluman, 2005).
To solve this arithmetic sequence problem, we are going to the formula “an=a1+ (n-1) d” to determine how much it will cost to construct a 90-foot tower. The n is considered the nth term which is 9 (1 for every 10 feet) we are looking for. The a1 is the first term; in this case it is the first $100 for the first 10 feet. The d is the common difference which is the $25 for each additional 10 feet, giving us a formula of
WEEK ONE
A9=100+ (9-1)25
= 100 + (8) (25)
= 100 + 200
= 300
The next step is to find the sum of the 9th term to determine the cost of the tower. For our next step we are going to use the formula sn = n (a1 + an) over 2 = 9 (100 + 300) over 2
= 9 (400) over 2
= 3600 ÷2
= 1800
It will cost $1800 to construct a 90-foot tower. This is a good example of using the arithmetic sequence for solving a real world situation. This is just one scenario that can be the basis of more difficult problems. The other form of sequence is the geometric sequence. The difference between arithmetic and geometric sequence is, you add the difference in the arithmetic and you multiply in the geometric to get the difference. After the first term has been identified, we multiply by a nonzero number; this number is our common ratio i.e 2, 4, 8, 16..., and 10, 100, 1000, 10000…, are just a couple of geometric sequences with 2 and 10 being their ratio. The general geometric sequence is written using the exponents of a1, a1r, a1r2, a1r3…. A good example of when we will use the geometric sequence is when we are dealing with money, especially if it is ours.
“A person deposits $500 in a savings account that pays 5% annual interest that is compounded yearly. At the end of 10 years, how much money will be in the savings account?”
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Money is always important to me especially since I do not have much. We will use the formula an = a1rn-1 to find our total 10 year savings. First we will consider the term of the loan which is an or10 years. Next is a1 which is the initial term of the loan being $500 or a1 is 500. Lastly is the ratio or difference of the loan with is 5%. Now, we have to take to account that, the first term already includes the first year’s interest to get the real first term. By doing so is the result of subtracting 1 to the ratio (rn-1) in the formula. an = a1rn-1 a10 = 525 (1.059) or a10 = 525 (1.05 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05) a10 = 525 (1.55) a10 = 813.75
After having $500 in the saving account for 10 years, there would be 813.75 saved. In the end, it will result in earnings of $313.75 in interest alone. It is interesting how finding the sequence of real numbers can help us in real life situations. If you want to build a project, knowing the arithmetic sequence will come in handy when needing to know how much it will cost. This will help you from getting the run around with contractors who are willing to take advantage of people who do not know how to calculate estimate of projects. It is the same when you are money in the bank. Knowing how much money you are supposed to have is always important. I do it because I am trying to save for my children’s college fund.
That is why it is important to remember the arithmetic formula when it comes to estimating project cost an=a1+ (n-1) d and sn = n (a1 + an) over 2. And when it comes to the geometric formula an = a1rn-1 when you are trying to put some money away for a retirement fund.
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I am looking forward to express more problems in my future.

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References
Bluman, A. G. (2005). Mathematics in our world (Ashford University Custom Edition). United States: McGraw-Hill.