Math Series and Sequences

Sequence

A function with a domain of natural (whole) numbers

The domain represents the positions of the terms of a sequence

The range is the set of terms in the sequence

Explicit Formula

Only uses n, the position of the term

Recursive Formula

Contains two part:

a. State initial term(s)

b. Write an equation for the nth term based on previous term(s)

Arithmetic Sequences

The difference between consecutive terms is constant, this is called the common difference: d = a_n-a₁

The Explicit Formula: a_n = a₁ - d(n-1)

Geometric Sequence

Has a common ratio, the ratio of consecutive terms is constant r = a_n/a_n-1 or r = \root(y-x)((a_(y))/(a_(x)))

Explicit Formula: a_n = a₁(r)^n-1

Series

the sum of the terms in a sequence

Arithmetic Series

sum of terms in a sequence

S_n = n/2(a₁ +a_n)

Geometric Series

S_n = a₁(1-rⁿ)/1-r

Infinite Geometric Series

S or S∞ means adding infinitely many terms

If absolute value of r is <1 and n → ∞, then rⁿ→0

The infinite series has a finite sum equal to: a₁/1-r

Arithmetic mean

a+b/2

Geometric Mean

The terms between two given terms of a geometric sequence

ⁿ√ab

Harmonic mean

the reciprocal of the average of the reciprocals of a and b

2/(1/a) + (1/b)

use to find the average of rates