General Mathematics: Functions, Annuities, and Sequences & Series

Comprehensive vocabulary flashcards covering function modeling, simple, general, and deferred annuities, and arithmetic and geometric sequences and series.

Piecewise Function

A function represented by multiple sub-functions over specific intervals of its domain, frequently applied to model real-life situations such as fare rates, tiered purchasing, and income tax brackets.

Annuity

A fixed sum of money paid or received at regular intervals, subject to a fixed compound interest rate (e.g., house rent, life insurance premiums, bond dividends, installment payments, and wages).

Simple Annuity

An annuity in which the payment interval coincides with or is exactly equal to the interest compounding period ($\text{Deposit Interval} = \text{Compounding Period}$).

General Annuity

An annuity where the payment dates do not coincide with or are not the same as the interest compounding interval ($\text{Deposit Interval} \neq \text{Compounding Period}$).

Deferred Annuity

An annuity whose periodic payments do not begin immediately, but instead start at a specified later date following a waiting period.

Ordinary Annuity

An annuity classification wherein periodic cash payments or deposits are made at the end of each payment interval.

Annuity Due

An annuity classification wherein periodic cash payments or deposits are made at the beginning of each payment interval.

Conversion Period Ratio ($b$)

The variable in general annuity formulas defined as $b = \frac{p}{c}$, where $p$ is the number of months in a payment interval and $c$ is the number of months in a compounding period.

Sequence

An ordered list of numbers arranged in a specific sequence or following a defined mathematical rule (e.g., $3, 7, 11, 15, \dots$).

Series

The mathematical expression representing the sum of the terms of a sequence (e.g., $3 + 7 + 11 + 15 + \dots$).

Arithmetic Sequence

A sequence of numbers where the difference between any two consecutive terms is constant, with the $n^{\text{th}}$ term given by $a_n = a_1 + (n - 1)d$.

Common Difference ($d$)

The constant numerical value added to any given term in an arithmetic sequence to produce the subsequent term.

Arithmetic Series

The indicated sum of the terms of an arithmetic sequence, evaluated using the formulas $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n - 1)d)$.

Geometric Sequence

An ordered set of numbers in which the quotient between any two consecutive terms is constant, with the $n^{\text{th}}$ term defined by $a_n = a_1 \cdot r^{n-1}$.

Common Ratio ($r$)

The constant factor by which each consecutive term in a geometric sequence or series is multiplied to obtain the next term.

Geometric Series

The sum of the terms of a geometric progression, with the finite sum calculated as $S_n = a_1\frac{1 - r^n}{1 - r}$ for $r \neq 1$.

Infinite Geometric Series

A geometric series with infinitely many terms that converges to the finite sum $S_\infty = \frac{a_1}{1 - r}$ if and only if the absolute value of the common ratio satisfies $|r| < 1$.

Sigma Notation

A compact mathematical notation using the Greek letter $\sum$ to represent the sum of a sequence, displaying the summation index, lower limit, upper limit, and the general term.