IB HL Honors Pre Calculus - Sequences, Series, and Binomial Expansions

Vocabulary flashcards covering topics in precalculus including sequences, series, multinomial expansion, Pascal's triangle, generalized binomial expansion, convergence, and historical connections.

Sequence

A set of numbers with a defined order, which can be understood as a discrete function $f: \mathbb{N} \rightarrow \mathbb{R}$ such that $f(n) = a_n$.

Arithmetic Sequence

A sequence in which there is a common difference $d$ between successive terms, defined explicitly by a linear function.

Geometric Sequence

A sequence in which there is a common ratio $r$ between successive terms, defined explicitly by an exponential function.

Common Difference

The constant value $d$ added or subtracted between successive terms in an arithmetic sequence ($d = a_n - a_{n-1}$).

Common Ratio

The constant factor $r$ multiplied between successive terms in a geometric sequence ($r = \frac{g_n}{g_{n-1}}$).

Recursive Formula

A definition for sequence terms based on preceding terms, requiring the initial term(s), an equation defining the next term, and the domain.

Explicit Formula

A formula that determines the value of term $a_n$ of a sequence solely based on its position $n$.

Trinomial Expansion Theorem

Theorem stating that the expansion of $(x + y + z)^n$ is the sum of all products $\frac{n!}{i!\,j!\,k!} x^i y^j z^k$ for non-negative integers $i, j, k$ such that $i + j + k = n$.

Multinomial Expansion Theorem

Theorem stating that $(x_1 + x_2 + \dots + x_m)^n$ expands into the sum of all possible products $\frac{n!}{i_1!\,i_2!\dots i_m!} x_1^{i_1} x_2^{i_2} \dots x_m^{i_m}$ where exponents $i_1, i_2, \dots, i_m$ sum to $n$.

Pascal's Triangle

A triangular arrangement of numbers in which each interior number is the sum of the two numbers directly above it, bounded by 1s on the outside edges.

Yang Hui Triangle

The name used in China for Pascal's Triangle, named after mathematician Yang Hui who published it in 1261, over 350 years before Pascal.

Al-Karaji

Persian mathematician (c. 1000 AD) who used the triangular arrangement of coefficients to study powers and algebra.

Combination

The number of arrangements of $n$ objects taken $r$ at a time where order does not matter, given by $\binom{n}{r} = \frac{n!}{(n-r)!\,r!}$.

Binomial Expansion Theorem

Theorem stating that for positive integer powers $n$, $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$.

Newton's Generalized Binomial Theorem

Extension of the Binomial Theorem by Isaac Newton allowing $(1 + x)^n$ to be expanded as an infinite polynomial series for any rational exponent $n \in \mathbb{Q}$ when $|x| < 1$.

Extension of the Binomial Theorem to Rational and Negative Indices

The formula $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \dots$ valid when $|x| < 1$ for any $n \in \mathbb{Q}$.

Convergent Infinite Sequence

An infinite sequence whose terms approach a specific value as $n \rightarrow \infty$.

Divergent Infinite Sequence

An infinite sequence whose terms do not approach a specific value as $n \rightarrow \infty$.

Convergent Infinite Series

An infinite series whose sum of terms approaches a specific finite value as $n \rightarrow \infty$.

Divergent Infinite Series

An infinite series whose sum of terms does not approach a specific finite value as $n \rightarrow \infty$.

Harmonic Series

The infinite series $\sum_{k=1}^\infty \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$, which diverges to infinity even though its individual terms converge to 0.

Infinite Geometric Series Sum Formula

The formula $S_\infty = \frac{g_1}{1 - r}$, representing the sum of a convergent infinite geometric series with first term $g_1$ and common ratio $|r| < 1$.

Zeno's Paradox

A philosophical problem posed by Zeno of Elea (c. 490–430 BC) questioning whether an infinite number of actions can produce a finite result, resolved using convergent infinite series.