Sequences

• Definition: A sequence is denoted as $am$, $a{m+1}$, $a{m+2}$, …, $an$, where:

  • Each $a_k$ (term) has:

  • $k$: subscript or index

  • $m$: subscript of initial term

  • $n$: subscript of final term (integer, $n \geq m$)

  • Notation indicates an infinite sequence.

Explicit Formulas

• An explicit formula provides a rule for how $a_k$ values depend on $k$.

• Example: Generating terms of sequences using explicit formulas.

Summation Notation

• Definition: If $m$ and $n$ are integers ($m \leq n$), summation notation is: \sum{k=m}^{n} ak = am + a{m+1} + … + a_n

  • $k$: index of summation, $m$: lower limit, $n$: upper limit.

Product Notation

• Uses Greek capital letter $\Pi$ to denote product of a sequence.

• Analogous to summation notation.

Properties of Summations and Products

• Theorem 5.1.1 properties:

  1. $\sum{k=m}^{n} ak + \sum{k=m}^{n} bk = \sum{k=m}^{n} (ak + b_k)$

  2. $\sum{k=m}^{n} c \cdot ak = c \cdot \sum{k=m}^{n} ak$

  3. Product properties similar to sum, including distributive law.

Recursive Definitions

• Factorial definition: $n! = n \cdot (n-1)!$, with $0! = 1$.

• Recursive formulas can simplify computations for sequences.

Fibonacci Sequence Example

• Defined via:

  1. $Fk = F{k-1} + F_{k-2}$ for $k \geq 2$

  2. Initial conditions: $F0 = 1$, $F1 = 1$.

• Calculation of subsequent terms: $F2 = 2$, $F3 = 3$, …, up to $F_{12} = 233$.

Compound Interest Example

• Recurrence relation for interest accumulation: $Pk = P{k-1} (1 + i/m)$ where $i$ is annual rate and $m$ is compounding frequency.

• Computation for specific interest scenarios (e.g., quarterly compounding) included.