QMS110 Lecture 10

Sequences

Definition and Notation

A sequence is a function whose domain is the set of natural (counting) numbers. Each term in the sequence is given by a function value at a specific integer.

• Finite sequence: Has a limited number of terms, often denoted as $a_1, a_2, ..., a_n$.

• Infinite sequence: Continues indefinitely, denoted as $a_1, a_2, a_3, ...$.

• General term: The $n$-th term is often written as $a_n$.

Example: The sequence $5, 10, 15, 20, 25, ...$ can be described by the formula $a_n = 5n$.

Ways to Define Sequences

• By specifying a general term (explicit formula)

• By listing the terms

• By a recursive formula (each term defined in terms of previous terms)

Finding Terms of a Sequence

To find specific terms, substitute the desired value of $n$ into the general formula.

nth term	First five terms	100th term
$(a)$ $2n-1$	$1, 3, 5, 7, 9$	$199$
$(b)$ $n^2-1$	$0, 3, 8, 15, 24$	$9999$
$(c)$ $\frac{n}{n+1}$	$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$	$\frac{100}{101}$
$(d)$ $\frac{(-1)^n}{2^n}$	$-\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, -\frac{1}{32}$	$\frac{1}{2^{100}}$

Summation Notation (Sigma Notation)

Definition

Summation notation uses the Greek letter $\Sigma$ to represent the sum of terms in a sequence:

• $\sum_{i=1}^{n} a_i$ means "the sum of $a_i$ from $i=1$ to $n$".

• The index of summation ($i$) indicates which terms to add.

Properties of Sums

• Linearity: $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$

• Constant multiple: $\sum_{i=1}^{n} c a_i = c \sum_{i=1}^{n} a_i$

• Difference: $\sum_{i=1}^{n} (a_i - b_i) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i$

Arithmetic Sequences

Definition and Properties

An arithmetic sequence is a sequence where the difference between consecutive terms is constant, called the common difference $d$.

• General form: $a_n = a_1 + (n-1)d$

• Each term: $a_{n+1} = a_n + d$

Examples

• Sequence: $5, -1, -7, -13, -19$ with $a_1 = 5$, $d = -6$

• Sequence: $13, 7, 1, -5, -11, -17$ with $a_1 = 13$, $d = -6$

Partial Sums of Arithmetic Sequences

The sum of the first $n$ terms (partial sum) is:

• $S_n = \frac{n}{2}(a_1 + a_n)$

• Alternatively, $S_n = \frac{n}{2}[2a_1 + (n-1)d]$

Applications

• Inventory: If a store starts with 90 cans and sells 3 per day, the inventory after $n$ days is $a_n = 90 - 3(n-1)$.

• Revenue: If daily revenue increases by $1000$each day, starting at $12000$, then $a_n = 12000 + (n-1)1000$.

Geometric Sequences

Definition and Properties

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio $r$.

• General form: $a_n = a_1 r^{n-1}$

• Each term: $a_{n+1} = r a_n$

Examples

• Sequence: $3, -6, 12, -24, 48$ with $a_1 = 3$, $r = -2$

• Sequence: $5, 15, 45, 135$ with $a_1 = 5$, $r = 3$

Partial Sums of Geometric Sequences

The sum of the first $n$ terms is:

• $S_n = a_1 \frac{1 - r^n}{1 - r}$, for $r \neq 1$

Applications

• Savings Account: If $125$is deposited at 5% interest compounded annually, after$n$years:$a_n = 125(1.05)^{n-1}$

• Future Value: For monthly deposits of $100$at 0.5% interest, the accumulated amount after$n$deposits:$S_n = 100 \frac{1 - (1.005)^n}{1 - 1.005}$

Infinite Geometric Series

Convergence and Sum

An infinite geometric series converges if $|r| < 1$ and diverges if $|r| \geq 1$.

• If convergent, the sum is $S = \frac{a_1}{1 - r}$

Examples and Applications

• Tax Rebate Multiplier: If each household spends 80% of a $1000$rebate, the total effect is$S = \frac{1000}{1 - 0.8} = 5000$

• Perpetuity: To provide $100,000$ per year forever at 2% interest, the required investment is $S = \frac{100,000}{1 - 1/1.02} = 5,100,000$

Summary Table: Arithmetic vs. Geometric Sequences

Type	General Term	Partial Sum Formula	Key Parameter
Arithmetic	$a_n = a_1 + (n-1)d$	$S_n = \frac{n}{2}(a_1 + a_n)$	Common difference $d$
Geometric	$a_n = a_1 r^{n-1}$	$S_n = a_1 \frac{1 - r^n}{1 - r}$	Common ratio $r$

Additional info:

• Recursive sequences are defined by relating each term to previous terms, e.g., Fibonacci sequence: $F_1 = 1, F_2 = 1, F_n = F_{n-1} + F_{n-2}$ for $n > 2$.

• Applications in business include modeling inventory, revenue growth, savings, and economic multipliers.