An annuity is a series of equal payments made at regular intervals over a specified period. It can be used to model various financial scenarios in algebra and sequences.
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Annuities can be classified as either ordinary annuities, where payments are made at the end of each period, or annuities due, where payments are made at the beginning of each period.
The future value of an annuity can be calculated using the formula: $$FV = P \left(\frac{(1 + r)^n - 1}{r}\right)$$ where $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the number of periods.
The present value of an annuity can be calculated using the formula: $$PV = P \left(\frac{1 - (1 + r)^{-n}}{r}\right)$$ where $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the number of periods.
In a geometric sequence context, each payment in an annuity can represent a term in the sequence.
Understanding how to manipulate these formulas involves knowledge of exponential functions and logarithms for solving equations.
Review Questions
What distinguishes an ordinary annuity from an annuity due?
How would you calculate the future value of an ordinary annuity with monthly payments?
What role does the interest rate play in determining both present and future values of an annuity?
Related terms
Geometric Sequence: A sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio.
Exponential Function: $f(x) = a \cdot b^x$, where $a$ is a constant and $b$ is a positive real number not equal to one; often used in financial calculations involving compound interest.
Present Value: $PV = \frac{FV}{(1 + r)^n}$; it represents today's value of a sum that will be received in the future given a specific interest rate.