Which of the Following are Geometric Sequences: A Comprehensive Guide for Businesses
In the world of business and finance, understanding geometric sequences is crucial for making informed decisions and maximizing profits. Unlock the power of geometric sequences today and take your business to the next level!
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant factor called the common ratio. In other words, if the first term is a and the common ratio is r, the sequence can be represented as follows:
a, ar, ar^2, ar^3, ...
It's important to note that the common ratio can be positive, negative, or even fractional.
Examples of Geometric Sequences
To illustrate, consider the following sequences:
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3, 6, 12, 24, ... has a common ratio of 2.
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8, -4, 2, -1, ... has a common ratio of -1/2.
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100, 50, 25, 12.5, ... has a common ratio of 1/2.
Identifying Geometric Sequences
To identify whether a sequence is geometric, simply calculate the ratio of consecutive terms. If the ratio is constant, then the sequence is geometric.
Table 1: Examples of Geometric Sequences
| Sequence |
Common Ratio |
| 3, 6, 12, 24, ... |
2 |
| 8, -4, 2, -1, ... |
-1/2 |
| 100, 50, 25, 12.5, ... |
1/2 |
Benefits of Using Geometric Sequences
Geometric sequences have numerous applications in business, including:
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Compound interest calculations: The formula for compound interest is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of compounding periods per year, and t is the number of years.
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Loan repayment schedules: The formula for calculating the monthly payment on a loan is M = P(r/n)/(1 - (1 + r/n)^(-nt)), where M is the monthly payment, P is the loan amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the loan term in years.
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Profit analysis: The formula for calculating the future value of an investment is FV = PV(1 + r)^n, where FV is the future value, PV is the present value, r is the annual interest rate, and n is the number of years.
How to Use Geometric Sequences
To use geometric sequences effectively, follow these steps:
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Identify the common ratio: Calculate the ratio of consecutive terms in the sequence.
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Use the formula: Once you know the common ratio, you can use the appropriate formula to solve for the unknown term.
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Apply to business problems: Utilize the formulas to calculate compound interest, loan payments, or future investment values.
Table 2: Applications of Geometric Sequences
| Application |
Formula |
| Compound interest |
A = P(1 + r/n)^(nt) |
| Loan repayment |
M = P(r/n)/(1 - (1 + r/n)^(-nt)) |
| Future investment value |
FV = PV(1 + r)^n |
Stories of Success
Story 1: Compounding Returns
Company A invested \$10,000 in a mutual fund with an annual interest rate of 10% compounded quarterly (n=4). After 5 years, the investment grew to approximately \$16,289.
How to Apply: Use the formula for compound interest (A = P(1 + r/n)^(nt)) to calculate the future value of the investment.
Story 2: Loan Repayment Optimization
Company B took out a loan of \$50,000 for 10 years with an annual interest rate of 6% compounded monthly (n=12). The monthly payment was approximately \$546.
How to Apply: Use the formula for loan repayment (M = P(r/n)/(1 - (1 + r/n)^(-nt))) to calculate the monthly payment.
Story 3: Investment Forecasting
Company C plans to invest \$200,000 in a real estate property that is expected to appreciate in value by 5% annually. In 10 years, the property is projected to be worth approximately \$320,714.
How to Apply: Use the formula for future investment value (FV = PV(1 + r)^n) to calculate the future value of the investment.
