Application of Exponential Functions & Sequences

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It describes growth or decay processes.

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What does it mean for a population to decline by 10% each year?

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A population declining by 10% each year means that each year, the population is reduced to 90% of its previous year's value.

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What is a geometric sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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How do you find the next term in a geometric sequence?

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To find the next term in a geometric sequence, multiply the current term by the common ratio.

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What is the formula for the nth term of a geometric sequence?

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The nth term of a geometric sequence can be found using the formula a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.

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What is an arithmetic sequence?

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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference.

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How do you find the nth term of an arithmetic sequence?

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The nth term of an arithmetic sequence can be calculated using the formula a_n = a_1 + d(n - 1), where a_1 is the first term, d is the common difference, and n is the term number.

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What is the common ratio in a geometric sequence?

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The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term.

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What is the common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence is the amount added to each term to get the next term.

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How do you calculate the value of an investment after a certain number of years with a percentage decrease?

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To calculate the value of an investment after a certain number of years with a percentage decrease, use the formula: final value = initial value * (1 - rate)^years.