Arithmetic and geometric sequences in tower construction and compound interest

¶ … 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 proceeding 100 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?

We see that there is a new price for every ten feet of tower. Each new price is $25 added to the previous price. Since repeated addition is involved, this is an arithmetic sequence. First, we need to identify the following numbers:

n = number of terms n = 9

d = the common difference

al = first term al = 100

an = last term an = a9

We know n = 9 because the tower increases in increments of ten feet, and the final height is 90 feet. 90/10=

To find the nth term of an arithmetic sequence, Page 271 of Mathematics in Our World gives us the following formula:

an = a1 + (n-1)d a9 = 100 + (9-1)

a9= 100 +8(25)

a9= 100 + 200

a9= 300

Now that we have found a9 is, we need to find the sum of the sequence is from a1 to a9. The formula is found in Mathematics in Our World on page 273.

Sn = the sum of n terms of the sequence (currently unknown)

a1 = the first term of the sequence an = the nth term of the sequence a1= 100

an= 300

Sn= n (a1+an)

Sn= 9(100 + 300)

Sn = 9(400)

Sn = 4.5(400) = 1800

It will cost $1,800 to build a 90-foot tower.

Question 37: 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 savings account?

Each year, 5% is added to the balance (B) in the savings account. This problem represents a Geometric sequence because each year is multiplied by the same number. In this case it is 1.05. It is 1 plus an additional 5% each year.

B + B (.05)

B (1+.05)

B (1.05)

First, we need to identify the following numbers:

n = the number of terms n = 10