Analysis, Finite Differences, and Pascal's Triangle Flashcards

Vocabulary flashcards covering finite differences, quadratic sequence formulas, odd triangle properties, Pascal's triangle identities, binomial expansion, and Fibonacci properties from the Analysis lecture series.

Method of Finite Differences

A technique used in analysis to determine the general term formula for a polynomial sequence, such as $a_n = An^2 + Bn + C$, by calculating successive differences between consecutive terms.

Quadratic Sequence Formula

The second-degree polynomial expression $a_n = An^2 + Bn + C$ used to model sequences that have a constant second difference.

Sum of First $n$ Odd Numbers

The cumulative total obtained by adding the first $n$ consecutive odd integers, defined by the formula $n^2$.

Row Sum of the Odd Number Triangle

The sum of all $n$ odd integers located in row $n$ of the odd number triangle, which is equal to $n^3$.

Sum of Top $n$ Rows in the Odd Number Triangle

The total sum of all terms across the first $n$ rows of the odd number triangle, calculated using the formula $\left(\frac{n(n+1)}{2}\right)^2$.

Pascal's Triangle

A triangular array of numbers where the entry in row $n$ at position $k$ corresponds to the binomial coefficient $\binom{n}{k}$.

Binomial Coefficient Symmetry

The symmetry property of Pascal's triangle expressed by the identity $\binom{n}{k} = \binom{n}{n-k}$.

Pascal's Identity

The additive relation $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$, showing that any interior entry in Pascal's triangle is the sum of the two entries directly above it.

Pascal's Triangle Row Sum Formula

The formula $\sum_{k=0}^n \binom{n}{k} = 2^n$, stating that the sum of all binomial coefficients in row $n$ equals $2^n$.

Binomial Expansion

The algebraic expansion of $(a+b)^n$ using coefficients from row $n$ of Pascal's triangle, such as $(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$.

Row Sum Doubling Property

The rule in Pascal's triangle stating that the sum of the numbers in row $n$ is exactly double the sum of the numbers in row $n-1$.

Fibonacci Recurrence Relation

The relation $F_n = F_{n-1} + F_{n-2}$, whose terms can be calculated by summing shallow diagonal paths in Pascal's triangle.