General Mathematics: Sequences, Series, Loans, and Mortgages

Vocabulary flashcards defining fundamental concepts in arithmetic sequences, geometric sequences and series, simple interest, loans, and mortgages.

Sequence

An ordered list of numbers that follows a certain pattern or rule.

Term (in a Sequence)

Each individual number that appears within a sequence.

Arithmetic Sequence

A sequence of numbers (also called an arithmetic progression) in which each term is obtained by adding a fixed constant called the common difference ($d$) to the previous term.

Common Difference ($d$)

The constant value added to each term in an arithmetic sequence to get the succeeding term, calculated as $d = a_n - a_{n-1}$.

$n\text{th}$ Term of an Arithmetic Sequence

The formula used to determine any term in an arithmetic sequence: $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

Arithmetic Mean

The value that comes exactly in the middle of two numbers in an arithmetic sequence, given by the formula $AM = \frac{a + b}{2}$.

Geometric Sequence

A sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant called the common ratio ($r$).

Common Ratio ($r$)

The constant multiplier between consecutive terms in a geometric sequence, calculated as $r = \frac{a_n}{a_{n-1}}$ or $r = \frac{a_{n+1}}{a_n}$.

$n\text{th}$ Term of a Geometric Sequence

The formula used to calculate any specified term in a geometric progression: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Geometric Mean

The middle term of a geometric sequence formed between two positive numbers $a$ and $b$, calculated as the square root of their product: $GM = \sqrt{a \cdot b}$.

Finite Geometric Series

The sum of a specific, limited number of terms of a geometric sequence, computed using the formula $S_n = \frac{a_1(1 - r^n)}{1 - r}$ where $r \neq 1$.

Interest

The cost of borrowing money paid to the lender, or the return earned on saved or invested money, typically expressed as a percentage of the principal.

Simple Interest ($I$)

Interest calculated solely on the principal amount using the formula $I = Prt$, where $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time period.

Ordinary Interest

A method of computing simple interest based on a $360$-day year or a $30$-day month.

Exact Interest

A method of computing simple interest based on the actual $365$-day calendar year.

Future Value ($F$)

The total accumulated amount of a loan or investment at the end of a term under simple interest, defined by $F = P + I$ or $F = P(1 + rt)$.

Loan

A financial agreement in which a borrower receives a sum of money (the principal) from a lender with the legal obligation to pay it back over time alongside interest.

Personal Loan

A type of loan taken out by individuals to finance personal expenses such as education, travel, or medical needs.

Business Loan

A loan obtained by a business entity to fund operations, purchase equipment, or cover starting and expansion costs.

Auto Loan

A specific type of secured loan designated for purchasing vehicles.

Mortgage

A long-term loan specifically used to buy real estate property (such as a house, building, or land) where the property itself acts as collateral.

Amortization

The process of gradually paying off a loan debt through regular, scheduled installment payments.

Collateral

The property or asset pledged by the borrower to the lender, which the lender has the right to repossess if the loan payments are not made.

Foreclosure

The legal proceeding by which a lender seizes and sells the mortgaged property when the borrower fails to make the required repayments.

Loan Date

The first day or starting date on which a loan is executed and funds are distributed to the borrower.

Maturity Date

The final due date of a loan on which the principal and all remaining interest must be completely settled.