Sequences, Series, and Sigma Notation

Review flashcards defining key rules and formulas for sigma notation, arithmetic sequences and series, and geometric sequences and series.

Distributive Property of Sigma Notation

The rule stating that a constant scalar factor can be factored outside of the summation: $\sum_{k=c}^n [d \cdot a_k] = d \sum_{k=c}^n [a_k]$.

Associative Property of Addition (Sigma Notation)

The rule stating that the sum or difference within a summation can be evaluated as separate summations: $\sum_{k=c}^n [a_k + b_k] = \sum_{k=c}^n [a_k] + \sum_{k=c}^n [b_k]$.

Constant Summation Formula

The formula for summing a constant $c$ across $n$ terms starting from $k=1$: $\sum_{k=1}^n [c] = c \cdot n$.

Sum of Consecutive Integers Formula

The summation formula for the first $n$ integers: $\sum_{k=1}^n [k] = \frac{n(n+1)}{2}$.

Sum of Consecutive Squares Formula

The summation formula for the squares of the first $n$ integers: $\sum_{k=1}^n [k^2] = \frac{n(n+1)(2n+1)}{6}$.

Sum of Consecutive Cubes Formula

The summation formula for the cubes of the first $n$ integers: $\sum_{k=1}^n [k^3] = \left[\frac{n(n+1)}{2}\right]^2$.

Arithmetic Sequence (Explicit Formula)

The explicit formula to find any term in an arithmetic sequence: $a_n = a_1 + d(n-1)$ or relative to term $k$ as $a_n = a_k + d(n-k)$.

Arithmetic Sequence (Recursive Formula)

A sequence formula where the first term $a_1$ is given, and each subsequent term is defined by $a_n = a_{n-1} + d$ for $n \ge 2$.

Arithmetic Sum ($S_n$)

The formula for the sum of the first $n$ terms of an arithmetic series: $S_n = \frac{n}{2}(a_1 + a_n)$.

Recursive Formula for Calc Section

A recursive model for amortized loans or balance changes given by $a_0 = B$ and $a_n = \left(1 + \frac{r}{k}\right)a_{n-1} - P$.

Geometric Sequence (Explicit Formula)

The explicit formula for the $n^{\text{th}}$ term of a geometric sequence: $a_n = a_1 r^{n-1}$ (also written as $g_n = g_0 r^n$ or $g_n = g_k r^{n-k}$).

Geometric Sequence (Recursive Formula)

A sequence formula where the first term $a_1$ is given, and each subsequent term is defined by $a_n = r \cdot a_{n-1}$ for $n \ge 2$.

Infinite Geometric Sum

The sum of an infinite geometric series when $|r| < 1$: $S = \frac{g_1}{1-r}$.