Like arithmetic sequences, geometric sequences can also be modeled using an explicit formula. Thus, Mary will have $6077.96 in her account after four years. To find how much Mary will have in her account after four years, we can use the recursive formula to find a_16 as follows: The recursive formula for this geometric sequence is: We can model Mary's savings pattern as a geometric sequence with a common ratio of r = 1 + (0.04/4) = 1.01, representing the interest rate for each quarter. The interest is compounded quarterly, and Mary wants to know how much she will have in her account after four years. Suppose Mary invests $5000 in a high-yield savings account that pays an annual interest rate of 4%. Compound interest is the interest that accumulates on a principal sum of money over time, and it is often calculated using the formula A = P(1 + r)^n, where A represents the final amount, P represents the principal amount, r represents the interest rate, and n represents the number of time periods. An example of a real-world context where geometric sequences can be used is the calculation of compound interest. Geometric sequences are patterns where each term is obtained by multiplying the previous term by a constant value known as the common ratio (r). Thus, the salary for the fifth month that we previously calculated using the recursive formula can also be obtained using the explicit formula as follows: Using the example above, we can write the explicit formula as follows: The explicit formula for an arithmetic sequence is given as: However, it is also possible to represent this same pattern using an explicit formula that directly calculates the nth term of the sequence. Thus, the employee will earn $2200 after five months of employment. To find the salary for the fifth month, we can use the recursive formula to find a_5 as follows: Starting from the initial salary of $2000, we can write recursive formula as follows: To calculate this, we can model the salary pattern as an arithmetic sequence with a common difference of $50. For instance, assume that an employer wants to determine how much an employee will earn after five months of employment if their salary is $2000 initially and is expected to increase by $50 every month. It is vital to know how to write these formulas in a real-world context because it can help solve practical problems and make accurate predictions about future outcomes.Īrithmetic sequences are patterns where each term is obtained by adding a constant value, known as the common difference (d), to the previous term. These sequences are often modeled using recursive and explicit formulas that give a straightforward representation of the patterns that govern them. Arithmetic and geometric sequences are an essential aspect of math, and they are widely used in the real-world setting.
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