Exponential Growth/Decay and Compound Interest

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What is exponential growth?

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Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to growth that accelerates over time. It can be represented by the formula y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is exponential decay?

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Exponential decay is the process of reducing an amount by a consistent percentage rate over a period of time. It can be represented by the formula y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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What is compound interest?

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Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It is calculated using the formula A = P(1 + r/n)^(nt), where 'A' is the amount of money accumulated after n years, including interest, 'P' is the principal amount, 'r' is the annual interest rate, 'n' is the number of times that interest is compounded per year, and 't' is the number of years.

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How do you calculate the total amount in a savings account with compound interest?

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To calculate the total amount in a savings account with compound interest, use the formula A = P(1 + r/n)^(nt), where 'A' is the total amount, 'P' is the principal, 'r' is the annual interest rate, 'n' is the number of times interest is compounded per year, and 't' is the number of years.

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What is the formula for calculating the future value of an investment with annual compounding?

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The future value of an investment with annual compounding is calculated using the formula A = P(1 + r)^t, where 'A' is the future value, 'P' is the principal amount, 'r' is the annual interest rate, and 't' is the number of years.

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What does it mean for a population to decrease by a certain percentage each year?

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When a population decreases by a certain percentage each year, it means that each year the population is reduced by that percentage of its current size. This can be modeled using the formula y = a(1 - r)^t, where 'a' is the initial population, 'r' is the decay rate, and 't' is time.

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How do you express a 30% decay in a mathematical equation?

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A 30% decay can be expressed in a mathematical equation as y = a(1 - 0.30)^t, where 'a' is the initial amount and 't' is time.

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What is the difference between simple interest and compound interest?

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Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal plus any accumulated interest. This means compound interest grows faster than simple interest over time.

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How do you find the total amount owed on a loan with compound interest?

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To find the total amount owed on a loan with compound interest, use the formula A = P(1 + r/n)^(nt), where 'A' is the total amount owed, 'P' is the principal loan amount, 'r' is the annual interest rate, 'n' is the number of compounding periods per year, and 't' is the number of years.

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What is the formula for calculating the balance of an investment after a certain number of years?

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The formula for calculating the balance of an investment after a certain number of years with compound interest is A = P(1 + r/n)^(nt), where 'A' is the balance, 'P' is the initial investment, 'r' is the interest rate, 'n' is the number of times interest is compounded per year, and 't' is the number of years.