Math_126_Week_One_Assignment

MATH 126 INTRODUCTION A sequence of number is a list of numbers that are related to each other by a specific rule. Each number in the sequence is called a term of the sequence. The two common types of sequences are the arithmetic sequences and the geometric sequences. Arithmetic sequence is a sequence of numbers in which each succeeding terms differs from the preceding term by the same amount. This amount is known as the common difference (p272). Geometric sequence is a sequence of terms in which each term after the first term is obtained by multiplying the preceding term by a nonzero number. This number is called the common ratio (p276). 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 than 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' Here is how it works: This problem involves arithmetic sequence since the labor cost for each successive ten feet remains constant at $25. The arithmetic sequences of the labor cost are: 10 feet charge = $100 20 feet charge = $125 30 feet charge = $150 40 feet charge = $175 50 feet charge = $200 60 feet charge = $225 70 feet charge = $250 80 feet charge = $275 90 feet charge = $300 To compute the “n” term using the formula from page 273 of Mathematics in Our World, in the pink box. First, is to identify the following given numbers: n = the number of all terms n = 9 d = the common difference d = 25 a1 = the first term a1 = 100 an = the last term an = a9 (to be computed) This is the formula to find the nth term of the sequence, or the 9th term in this problem. an = a1 + (n - 1) d a9 = 100 + (9 – 1) 25 a9 = 100 + (8) (25) a9 = 100 + 200 a9 = 300 The amount of last term is a9 = $300. Now, to find the sum of the sequence from a1 to a9. Use the formula from page 275 of Mathematics in Our world, right below the blue “Try This One” box. Sn = n (a1 + a9) 2 Sn = 9 (100 + 300) 2 Sn = 9 (400) 2 Sn = 3200 2 Sn = 1800 Therefore, the cost for building a 90 foot tower is $1800. A person deposited $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 saving account' Here is how it works: After one year the savings will earn 5% of $500 as interest, or $25, so the new balance for the second year is $525. If B = the balance, it would look like this: B + (.05) B B (1 + .05) B (1.03) For each year the balance will be multiplied by 1.05. The repeated multiplication by the same number shows that this is a geometric sequence. To start the computation we need to identify the following given numbers: n = the number of terms n = 10 r = the common ratio r = 1.05 a1 = the first term a1 = 525 the balance at the end of first year Now to find out what is the balance at the end of 10 years, use the formula from page 277 of Mathematics in Our World, in the blue box. an = a1(rn-1) a10 = 525(1.059) a10 = 525(1.551328…) a10 = 814.45 Therefore, the new balance at the end of 10 years is $814.45. References Bluman: Mathematics in Our World. By the McGraw−Hill Companies Copyright ©2008, http://www.primisonline.com