PRECALCULUS - Definitions and Formulas

Formula for an arithmetic sequence

aₙ = a₁ + (n - 1)d

2 Formulas for finding the nth term of an arithmetic sequence

aₙ = dn + c and c = a₁ - d

Formula for the finite sum of an arithmetic series (with d)

Sₙ = n/2 [2a₁ + (n-1)d]

Formula for the finite sum of an arithmetic series (with aₙ)

Sn = n/2 (a₁ + aₙ)

System of equations

A set of two or more equations with the same variables. A solution is a set of values that satisfies all equations simultaneously.

Non-linear system

A system in which at least one equation is not linear.

Elimination

transform one or both equations so that you can eliminate one of the variables by combining the equations together

Substitution

solve one equation for one variable and substitute it to the other equation(s).

Graphing

useful for visual understanding

Sequence

It is a list of items in a specific order. This can be classified by the number of terms: infinite and finite, or by pattern: arithmetic, geometric, harmonic, and fibonacci.

Infinite Sequence

It is a sequence that contains a countless number of terms and an ellipsis (...)

Finite Sequence

It is a sequence that contains a limited number of terms. This means that the last term of the sequence is known.

Series

The sum of the terms of a sequence.

Arithmetic sequence

It is a sequence of numbers in which each term after the first is obtained by adding a constant number. This constant is called the common difference.

Arithmetic series

It is the total obtained by adding all the terms of an arithmetic sequence, where each term increases or decreases by a constant difference.

Geometric sequence

It is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a constant number. This constant is called the common ratio.

Geometric series

it is the total obtained by adding all the terms of a geometric sequence, where each term is multiplied by a constant ratio to get the next term.