with clear illustrations and examples explain how arithmetic and geometric progression can be used to solve problems related to
a)construction of business
b)banking business
c)production/operation function
well, in all of them interest, present value, future value, and inflation costs are always being used.
a) Construction of Business:
Arithmetic Progression (AP):
Arithmetic progression can be used in the construction business to calculate the cost of materials or expenses over time. Let's consider an example of a construction project where the cost of materials increased by $500 per month.
In this case, we have an arithmetic progression, where the common difference (d) is $500. We can use the formula for the nth term of an AP to calculate the cost at a specific time:
nth term (An) = a + (n - 1) * d
Where 'a' is the first term and 'n' is the term number. For example, if the cost of materials at the beginning of the project (a) is $2,000, we can calculate the cost after 5 months as follows:
A5 = 2000 + (5 - 1) * 500
= 2000 + 4 * 500
= 2000 + 2000
= $4000
So, after 5 months, the cost of materials would be $4000.
Geometric Progression (GP):
Geometric progression can be used in the construction business to calculate exponential growth or decay. For example, let's say the construction company's revenue is growing at a rate of 10% per quarter.
In this case, we have a geometric progression, where the common ratio (r) is 1 + (10%/100%) = 1.1. We can use the formula for the nth term of a GP to calculate the revenue at a specific time:
nth term (An) = a * r^(n-1)
Where 'a' is the first term and 'n' is the term number. For instance, if the initial revenue (a) is $100,000, we can calculate the revenue after 3 quarters as follows:
A3 = 100000 * 1.1^(3-1)
= 100000 * 1.1^2
= 100000 * 1.21
= $121,000
Thus, after 3 quarters, the revenue of the construction company would be $121,000.
b) Banking Business:
Arithmetic Progression (AP):
Arithmetic progression can be used in the banking business to calculate the interest amount or loan repayment amounts over time. Let's consider an example where a customer takes a loan with a monthly repayment amount of $500 and an interest rate of 7% per annum.
In this case, we have an arithmetic progression for the loan repayment, where the common difference (d) is $500. We could use the formula for the nth term of an AP to determine the principal amount after a certain number of months:
nth term (An) = a + (n - 1) * d
Where 'a' is the first term (loan amount) and 'n' is the term number. For instance, if the loan amount is $10,000, and we want to determine the principal amount after 12 months:
A12 = 10000 + (12 - 1) * 500
= 10000 + 11 * 500
= 10000 + 5500
= $15,500
So, after 12 months, the principal amount would be $15,500.
Geometric Progression (GP):
Geometric progression can be used in the banking business to calculate compound interest. Let's assume a customer places $10,000 in a savings account with an annual interest rate of 5%.
In this case, we have a geometric progression for the interest growth, with a common ratio (r) of 1 + (5%/100%) = 1.05. We could use the formula for the nth term of a GP to calculate the balance after a specific time:
nth term (An) = a * r^(n-1)
Where 'a' is the initial amount and 'n' is the term number. For example, if the initial deposit is $10,000, and we want to determine the balance after 3 years:
A3 = 10000 * 1.05^(3-1)
= 10000 * 1.05^2
= 10000 * 1.1025
= $11,025
So, after 3 years, the balance of the savings account would be $11,025.
c) Production/Operation Function:
Arithmetic Progression (AP):
Arithmetic progression can be used in production or operations to analyze linear growth or cost increases. For example, let's consider a manufacturing company that experiences monthly production growth of 200 units.
In this case, we have an arithmetic progression, with the common difference (d) equal to 200. We could use the formula for the nth term of an AP to calculate the production level at a specific time:
nth term (An) = a + (n-1) * d
Where 'a' is the initial production level and 'n' is the term number. For instance, if the company starts with a production level of 1000 units, and we want to determine the production after 6 months:
A6 = 1000 + (6-1) * 200
= 1000 + 5 * 200
= 1000 + 1000
= 2000 units
Thus, after 6 months, the production would be 2000 units.
Geometric Progression (GP):
Geometric progression can be used in production or operations to analyze exponential growth or decay. Let's assume a company's production capacity grows by 10% annually.
In this case, we have a geometric progression, with the common ratio (r) equal to 1 + (10%/100%) = 1.10. We can use the formula for the nth term of a GP to calculate the production capacity at a specific time:
nth term (An) = a * r^(n-1)
Where 'a' is the initial production capacity and 'n' is the term number. For example, if the company starts with a production capacity of 100 units, and we want to determine the capacity after 4 years:
A4 = 100 * 1.10^(4-1)
= 100 * 1.10^3
= 100 * 1.331
= 133.1 units (rounded to one decimal place)
Therefore, after 4 years, the production capacity would be approximately 133.1 units.