2.4 sequences and summations

Fibonacci sequence

The Fibonacci sequence, f0, f1, f2, ... , is defined by the initial conditions f0 = 0, f1 = 1, and the recurrence relation fn = fn−1 + fn−2 for n = 2, 3, 4, ... .

sequence

a function from a subset of the set of integers to a set S.

geometric progression

a sequence of the form a, ar, ar^2, ..., ar^n. where the initial term a and common ratio r are real numbers

recurrence relation

an expression that expresses a(n) in terms of one or more of the previous terms of the sequence, namely a0, a1, ..., a(n-1), for all integers n with n >= n(0), where n(0) is a non-negative number.

iteration

apply a recurrence relation multiple times to reach a closed formula

we use the notation a(n) to ---

denote the image of the integer n. we call a(n) a term of the sequence

a geometric progression is a discrete analogue of the exponential function ---

f(x) = ar^x

an arithmetic function is a discrete analogue of the linear function ---

f(x) = dx + a

a sequence is called a solution of a recurrence relation if ---

its terms satisfy the recurrence relation

arithmetic progression

sequence of the form a, a+d, a+2d, ..., a+(n)d

initial conditions

specify the terms that precede the first terms where the recurrence relation takes effect

Forward Substitution

successive terms beginning with the initial condition and ending with an

backward substitution

we began with a(n) and iterated to express it in terms of falling terms of the sequence until we found it in terms of a1

we say that we have solved the recurrence relation together with the initial conditions when ---

we find an explicit formula, called a closed formula, for the terms of the squence