Section 8.1: Defining and Using Sequences and Series

Section 8.1 Fundamentals of Sequences

Conceptual Definition of Sequences

A sequence is defined as an ordered list of numbers. Each number in the sequence is referred to as a term. Sequences are represented mathematically as identifying the position of a term relative to the whole set:

$a_1, a_2, a_3, a_4, \dots, a_n, \dots$

Finite and Infinite Sequences

1. Finite Sequence: A function that has a limited number of terms. Its domain is the finite set of integers ${1, 2, 3, \dots, n}$.    - Example: $2, 4, 6, 8$

2. Infinite Sequence: A function that continues without stopping. Its domain is the set of all positive integers.    - Example: $2, 4, 6, 8, \dots$

Mathematical Notation and Domain

• The domain of a sequence typically starts with $n=1$. However, the domain may begin with $0$ if specified.

• If starting with $0$, the domain for a finite sequence is ${0, 1, 2, 3, \dots, n}$ and for an infinite sequence is the set of nonnegative integers.

• The values in the range are the actual terms of the sequence ($a_n$).

• A sequence can be specified by an equation or rule, such as $a_n = 2n$ or $f(n) = 2n$.

Writing Rules and Terms of Sequences

Generating Terms from a Rule

To find the terms of a sequence given its rule, substitute the values of the domain ($n = 1, 2, 3, \dots$) into the equation.

Example 1: Generating Terms

• For $a_n = 2n + 5$:   - $a_1 = 2(1) + 5 = 7$   - $a_2 = 2(2) + 5 = 9$   - $a_3 = 2(3) + 5 = 11$   - $a_4 = 2(4) + 5 = 13$   - $a_5 = 2(5) + 5 = 15$   - $a_6 = 2(6) + 5 = 17$

• For $f(n) = (-3)^{n-1}$:   - $f(1) = (-3)^{1-1} = 1$   - $f(2) = (-3)^{2-1} = -3$   - $f(3) = (-3)^{3-1} = 9$   - $f(4) = (-3)^{4-1} = -27$   - $f(5) = (-3)^{5-1} = 81$   - $f(6) = (-3)^{6-1} = -243$

Deducing Rules from Patterns

When given a set of terms, you must observe the relationship between the position of the term ($n$) and the value ($a_n$).

Pattern Analysis Examples:

1. Sequence: $-1, -8, -27, -64, \dots$     - Pattern: These are the cubes of negative integers: $(-1)^3, (-2)^3, (-3)^3, (-4)^3$     - Next Term: $a_5 = (-5)^3 = -125$     - Rule: $a_n = (-n)^3$

2. Sequence: $0, 2, 6, 12, \dots$     - Pattern: Term values follow $0(1), 1(2), 2(3), 3(4)$.     - Next Term: $f(5) = 4(5) = 20$     - Rule: $f(n) = (n-1)n$

Note on Rule Uniqueness: There may be more than one rule for a given list of terms. For example, the list $2, 4, 8, \dots$ could follow $a_n = 2^n$ or $a_n = n^2 - n + 2$.

Graphing Sequences

• To graph a sequence, let the horizontal axis represent the position numbers ($n$, the domain) and the vertical axis represent the terms ($a_n$, the range).

• Important Constraint: The points of a sequence graph are discrete. You must not draw a continuous curve/line connecting the points because the sequence is defined only for integer values of $n$.

Series and Summation Notation

Definition of a Series

When the terms of a sequence are added together, the resulting expression is called a series.

• Finite Series: An addition of a specific number of terms. Example: $2 + 4 + 6 + 8$

• Infinite Series: An addition that continues indefinitely. Example: $2 + 4 + 6 + 8 + \dots$

Summation/Sigma Notation

Summation notation uses the uppercase Greek letter sigma ($\sum$) to express a series concisely. It identifies the index of summation, the lower limit, and the upper limit.

Reading Summation: $\sum_{i=1}^{4} 2i$ is read as "the sum of $2i$ for values of $i$ from $1$ to $4$."

Examples of Writing Series in Summation Notation:

1. Series: $25 + 50 + 75 + \dots + 250$     - Each term is $25i$.     - Lower limit: $i = 1$ ($25 \times 1 = 25$).     - Upper limit: $i = 10$ ($25 \times 10 = 250$).     - Notation: $\sum_{i=1}^{10} 25i$

2. Series: $\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \dots$     - Pattern: The denominator is $1$ more than the numerator: $a_i = \frac{i}{i+1}$.     - Upper limit: Infinity (indicated by the ellipsis).     - Notation: $\sum_{i=1}^{\infty} \frac{i}{i+1}$

Evaluating a Sum

To find the sum of a specific series, substitute each value of the index into the expression and add the results.

• Example: Find $\sum_{k=4}^{8} (3 + k^2)$   - Terms: $(3 + 4^2) + (3 + 5^2) + (3 + 6^2) + (3 + 7^2) + (3 + 8^2)$   - Values: $19 + 28 + 39 + 52 + 67 = 205$

Special Summation Formulas

For series with many terms, explicit addition becomes tedious. The following formulas are used to find the sums of specific types of sequences:

1. Sum of $n$ terms of $1$:    $\sum_{i=1}^{n} 1 = n$

2. Sum of first $n$ positive integers:    $\sum_{i=1}^{n} i = \frac{n(n + 1)}{2}$

3. Sum of squares of first $n$ positive integers:    $\sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6}$

Real-Life Applications

Example Case: Pyramid of Apples

Scenario: Stacking apples in a square pyramid with 7 layers.

• Layer Rule: The number of apples in layer $n$ is given by $a_n = n^2$.

• Calculating Total: To find the total apples in the stack, find the sum from layer 1 to 7:   $\sum_{i=1}^{7} i^2 = \frac{7(7 + 1)(2 \times 7 + 1)}{6} = \frac{7(8)(15)}{6} = 140$

• Verification: If the stack had 9 layers, the total would follow the same formula with $n = 9$.

Example Case: Salary Progression

Scenario: Starting salary of $33,000 with annual raises of $2,400.

• Rule for salary in year $n$: $a_n = 33000 + 2400(n - 1)$.

Example Case: Savings Goal

Scenario: Saving 1 penny on day 1, 2 pennies on day 2, and so on ($a_n = n$).

• Total Saved after 100 days:   \sum_{i=1}^{100} i = \frac{100(100 + 1)}{2} = 5050 \text{ pennies} = $50.50

• Goal Analysis: To save $500 ($50,000$ pennies), one must solve $\frac{n(n+1)}{2} = 50000$ for $n$.

Example Case: Regular Polygons

Interior Angle Measure ($a_n$): For a regular n-sided polygon ($n \ge 3$): $a_n = \frac{180(n - 2)}{n}$

• For the Guggenheim Museum skylight (a dodecagon where $n = 12$):   - Total sum of angles $T_n = \sum_{i=1}^{n} a_i$ is simplified by the geometry rule $T_n = 180(n - 2)$.   - Substitution for $n = 12$: $T_{12} = 180(12 - 2) = 180(10) = 1800^{\circ}$.

Example Case: Tower of Hanoi

Scenario: Moving $n$ rings from one peg to another.

• Minimum moves required for $n$ rings: $1, 3, 7, 15, 31, \dots$

• Sequence rule: $a_n = 2^n - 1$

• Minimun moves for 8 rings: $2^8 - 1 = 256 - 1 = 255$

Questions & Discussion

Critical Reasoning

• Writing rules: A rule for the nth term can be found by mapping domain inputs ($1, 2, 3 \dots$) to the range and looking for common increments (arithmetic) or multipliers (geometric).

• Relationships:   - (a) Arithmetic sequence: The difference between consecutive terms is constant.   - (b) Geometric sequence: The ratio between consecutive terms is constant.

• Index manipulation: If a sum begins at a value higher than 1 (e.g., $\sum_{i=3}^{1659} i$), friend claims to use the formula. This is achieved by calculating the sum from 1 to 1659 and subtracting the sum from 1 to 2.

Common Errors to Avoid

• Formula misuse: The summation formulas (like $\frac{n(n+1)}{2}$) apply only when the lower limit of summation is $1$. If the index starts at $i=2$ or higher, the formula cannot be used directly on the upper limit without adjustments.

• Graphing mistake: Do not connect the dots when plotting sequences. Sequences are discrete functions.

• Summation Evaluation: Ensure you evaluate the expression for the first index value as part of the total. Failing to include the lower bound term is a common calculation error.