5.3 Sequences, Series, & Factorials

5.3 Sequences, Series, & Factorials

In college algebra, discrete mathematics provides essential analytical tools for modeling patterns, financial growth, and series approximations. A sequence is an ordered function mapping natural numbers to real numbers, while a series represents the sum of a sequence's terms. Mastering sequences requires distinguishing arithmetic and geometric structures and working fluently with factorial notation.

---

Foundations of Sequences and Series

Core Definitions

• Sequence ($a_n$): An ordered list of numbers $a_1, a_2, a_3, \dots, a_n, \dots$ where $a_n$ denotes the $n$-th term.
• Explicit Formula: Expresses $a_n$ directly as a function of term position index $n$ (e.g., $a_n = 3n + 1$).
• Recursive Formula: Expresses $a_n$ using preceding terms (e.g., $a_n = a_{n-1} + 3$ with seed $a_1 = 4$).
• Series ($S_n$): The sum of the terms of a sequence up to position $n$:

$S_n = \sum_{k=1}^n a_k = a_1 + a_2 + \dots + a_n$

---

Arithmetic Sequences & Partial Sums

An arithmetic sequence is a sequence in which the difference between consecutive terms is a constant scalar called the common difference ($d$).

$d = a_n - a_{n-1} \quad \text{for all } n \ge 2$

Key Formulas for Arithmetic Sequences

1. Explicit $n$-th Term Formula: $a_n = a_1 + (n - 1)d$
2. Finite Partial Sum Formula ($S_n$): $S_n = \frac{n}{2}\left(a_1 + a_n\right) \quad \text{or} \quad S_n = \frac{n}{2}\left[ 2a_1 + (n - 1)d \right]$

Intuition: Gauss's partial sum method pairs the first and last terms ($a_1 + a_n$), second and second-to-last terms, producing $\frac{n}{2}$ pairs of equal sum.

---

Geometric Sequences & Infinite Series Convergence

A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a non-zero constant called the common ratio ($r$).

$r = \frac{a_n}{a_{n-1}} \quad \text{for all } n \ge 2$

Key Formulas for Geometric Sequences

1. Explicit $n$-th Term Formula: $a_n = a_1 \cdot r^{n-1}$
2. Finite Partial Sum Formula ($S_n$): $S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)$
3. Infinite Geometric Series Sum ($S_\infty$): An infinite geometric series $\sum_{k=1}^\infty a_1 r^{k-1} = a_1 + a_1 r + a_1 r^2 + \dots$ converges to a finite sum if and only if $|r| < 1$: $S_\infty = \frac{a_1}{1 - r} \quad \text{for } |r| < 1$ If $|r| \ge 1$, the infinite series diverges (the sum grows without bound or oscillates infinitely).

Summary Comparison Table

---

Factorial Notation & Algebraic Simplifications

For any positive integer $n$, the factorial function $n!$ (read "$n$ factorial") is defined as the product of all positive integers less than or equal to $n$:

$n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$

Fundamental Properties & Conventions

• Zero Factorial: By mathematical convention, $0! = 1$ (which preserves consistency in combinatorial formulas).
• Recursive Identity: For $n \ge 1$, $n! = n \cdot (n - 1)!$

Algebraic Factorial Simplification

When simplifying quotients of factorials containing variables, expand the larger factorial down to match the smaller factorial to cancel common factors:

$\frac{(n + 2)!}{n!} = \frac{(n + 2)(n + 1)n!}{n!} = (n + 2)(n + 1) = n^2 + 3n + 2$

---

Comprehensive Worked Examples

Worked Example 1: Arithmetic Sequence & Sum Application

Problem: An auditorium has $18$ seats in the first row, $22$ seats in the second row, $26$ seats in the third row, and continues to increase by $4$ seats per row. How many total seats are in the auditorium if there are $25$ rows?

Step-by-Step Solution:

1. Identify sequence parameters:
   • First term $a_1 = 18$.
   • Common difference $d = 22 - 18 = 4$.
   • Number of terms $n = 25$.
2. Find the 25th term $a_{25}$ using $a_n = a_1 + (n - 1)d$: $a_{25} = 18 + (25 - 1)(4) = 18 + 24(4) = 18 + 96 = 114$
3. Calculate total sum $S_{25}$ using $S_n = \frac{n}{2}(a_1 + a_n)$: $S_{25} = \frac{25}{2}(18 + 114) = 12.5 \times 132 = 1650$ Conclusion: There are $1650$ total seats.

---

Worked Example 2: Geometric Sequence & Infinite Sum

Problem: An infinite geometric series has a first term $a_1 = 45$ and an infinite sum $S_\infty = 75$. Find the common ratio $r$ and the third term $a_3$.

Step-by-Step Solution:

1. Use infinite sum formula $S_\infty = \frac{a_1}{1 - r}$ to solve for $r$: $75 = \frac{45}{1 - r}$
2. Cross-multiply and isolate $r$: $75(1 - r) = 45 \implies 1 - r = \frac{45}{75} = \frac{3}{5}$ $r = 1 - \frac{3}{5} = \frac{2}{5}$ (Note: $|r| = 0.4 < 1$, verifying convergence).
3. Calculate third term $a_3 = a_1 r^2$: $a_3 = 45 \left(\frac{2}{5}\right)^2 = 45 \cdot \frac{4}{25} = \frac{180}{25} = \frac{36}{5} = 7.2$

---

Worked Example 3: Simplifying Algebraic Factorials

Problem: Simplify the algebraic expression for any integer $n \ge 2$: $E = \frac{(n + 2)!}{n!} - \frac{(n + 1)!}{(n - 1)!}$

Step-by-Step Solution:

1. Expand first quotient: $\frac{(n + 2)!}{n!} = \frac{(n + 2)(n + 1)n!}{n!} = (n + 2)(n + 1) = n^2 + 3n + 2$
2. Expand second quotient: $\frac{(n + 1)!}{(n - 1)!} = \frac{(n + 1)(n)(n - 1)!}{(n - 1)!} = (n + 1)n = n^2 + n$
3. Subtract the expanded expressions: $E = (n^2 + 3n + 2) - (n^2 + n) = n^2 - n^2 + 3n - n + 2 = 2n + 2$