# Sequence

A sequence is basically a general list type mathematical tool that can be indexed. It's technically a function where the domain is a consecutive set of integers.

g: {1,2,3,4} -> R

g(1) = 3.98, g(2) = 4.0 etc...

Sequences can have positive or negative indexes.

They can be infinite or finite

For a sequence to be considered increasing or decreasing, every indices must be larger or smaller than the previous one, respectfully.

# Formula

Sequences can be defined as a formula:

d(k) = 2^k where k = 0,1,2,3...

# Geometric sequence

A sequence of real numbers where each element is obtained by multiplying the previous element by some common ratio.

a, ar, ar^2, ar^3, ar^4,...,ar^n

Compound interest is a geometric sequence

base_amount(1+rate)^years

# Arithmetic sequence

A sequence of real numbers where each element is obtained by adding a constant to the previous element.

a, a+d, a+2d,..., a+nd

# Summation notation

t (upper limit)
∑ a[i]              = a[s] + a[s + 1] + a[s + 2] + ... + a[t]
i = s (lower limit)

Sums can also sometimes be expressed without summation notation, this is called closed form:

Sum of terms in a geometric sequence for any real number r != 1 and any integer
n >= 1:

n-1
∑ (a*r^k) = [a(r^n - 1)]/(r - 1)
k=0

# Recurrence relation

When you cannot create a general formula for a sequence and need to refer to the previous terms of the sequence

example: fibonacci sequence

f[0] = 0
f[1] = 1
f[n] = f[n-1] + f[n-2] for n >= 2

This is an example of a dynamical system where state changes over time and a discrete time dynamical system where time is divided into intervals and the state of the system is fixed during each interval and is a function of the previous time interval.

# Limits

The limit of an infinite sequence is just where the sequence approaches as you use sequence elements close to the limit. Just think of it as a function of n. If you do not see a clear single grouping, the limit is not defined.