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Sequence

A set of numbers arranged in a special order or pattern.

Slope is the first difference (m)

How do you find create a formula using first differences (linear)?

Half of the second difference is the a-value

How do you create a formula with second differences (quadratic)?

Fibonacci sequence

Each number, after the first two numbers is the sum of the previous two numbers

Arithmetic sequences

The common difference d, between each consecutive term is constant; each term is formed by adding a fixed quantity to the term before it

Recursion formula

A formula in which each term of a sequence is generated from the preceding term or terms

tn=tn-1 - x

What is the general recursion formula?

Pascals triangle

An arrangement of numbers. Each row is generated by calculating the sum of pairs of consecutive terms in the previous row. Row 0 has only a 1.

tn,m where

Positions of terms in pascals triangle

tn-1,m-1 , tn-1,m

What is the sum of tn,m?

Common difference (d)

Corresponds to the difference/rate of change between two consecutive terms in an arithmetic sequence

Arithmetic series

Sum of the terms of a sequence

Partition

Let A be a set. A partition of (or on) A is a set of nonempty, pairwise disjoint (no element is common between two parts) sets whose union is A.

Binomial coefficient

Let n, k be integers. (n / k) denotes the number of k-element subsets of an n-element set. "n choose k (things)" (n / k) = n!/k!(n-k)!

Pascal's Identity

(n / k) = (n-1 / k-1) + (n-1 / k)

Proof by Contrapositive

A direct proof of an implication's contrapositive. This uses the fact that for every two statements P and Q, the implication P → Q and its contrapositive are logically equivalent.

Proof by contradiction

To prove "if A, then B," assume A and not B - look for a contradiction that then proves B. (Good for proving an empty set or uniqueness)

function

If you repeat an INPUT, and you always get the same OUTPUT, your data is a ______________________ .

Domain, Image

The domain is the set of all possible first elements of the ordered pairs in f - dom f. The image is the set of the second elements - im f.

One-to-one

A function f is called one-to-one provided that, whenever (x, b), (y, b) ∈ f, we must have x=y. In other words, if x≠y, then f(x)≠f(y). (Has to pass the horizontal line test). Inverse func…

Onto

Let f: A → B. f is onto B provided that for every b in B, there is an a in A so that f(a)=b. in other words, im f=B.

Bijection

The function is both one-to-one and onto.

(a,a) for every a in the set

Ex. Sn = t1 + t2 + ... + tn-1 + tnReflexive

Symmetric

Flip the left side of the equation to the right side.

Antisymmetric

(a,b) and (b,a) ONLY if a=b

Asymetric

(a,b) implies NOT(b,a)

Transitive

(a,b) and (b,c) implies (a,c)

Reflexive

(a,a) for every a in the set