Arithmetic and Geometric Series Study Guide

Fundamental Concepts of Series

• A series is defined as the sum of the terms in a sequence.
• The mathematical representation for the sum of the first $n$ terms of a series is denoted as $S_n$:
• $S_n = t_1 + t_2 + t_3 + t_4 + \dots + t_n$

Arithmetic Series

• An arithmetic series is the sum of terms in an arithmetic sequence.
• Example Comparison:
  • Arithmetic sequence: $1, 3, 5, 7, \dots$
  • Arithmetic series: $1+3+5+7+ \dots$
  • Illustrative calculation: In the sequence above, the sum of the first four terms ($S_4$) is $16$.

Formulas for Determining the Sum of an Arithmetic Series

In general, the sum of the first $n$ terms in an arithmetic series can be found using one of two formulas depending on the known variables:

1. Using the First and Last Terms

This formula is used when you are given the first term ($a$) and the last term ($t_n$) in the sequence:

• Formula: $S_n = \frac{n}{2}(a + t_n)$
• Variables:
  • $a$: The first term of the series.
  • $n$: The number of terms or the term number.
  • $t_n$: The last term in the series.
  • $S_n$: The sum of the first $n$ terms.

2. Using First Term, Common Difference, and Number of Terms

This formula is used when you are given the first term ($a$), the common difference ($d$), and the number of terms ($n$):

• Formula: $S_n = \frac{n}{2}[2a + (n - 1)d]$
• Variables:
  • $d$: The common difference between terms.

Arithmetic Series Examples and Applications

Example 1: Determining Term Count and Focused Sum

Given the arithmetic series: $-7, -11, -15, \dots, -167$

• Part A: Determine the total number of terms ($n$) in the series.
• Part B: Determine the sum of the first $30$ terms in the series.

Example 2: Financial Application - Salary and Raises

After graduating from university, Ryan accepts a position as an accountant for a small firm.

• Salary Data:
  • First-year salary ($a$): $\$48,500$
  • 12th-year salary ($t_{12}$): $\$73,250$
• Conditions: Ryan receives a fixed annual raise, meaning his salary progression forms an arithmetic sequence with $12$ terms.
• Part A: Calculate the fixed annual raise ($d$) Ryan receives.
• Part B: Calculate Ryan's total earnings over the full $12$ years of his employment ($S_{12}$).

Practice Problems for Arithmetic Series

• Summation Practice: Find the sum of the first $18$ terms for the arithmetic series: $2.5 + 4 + 5.5 + \dots$
  • Answer: $274.5$
• Real-world Application - Theatre Seating:
  • An auditorium contains $20$ seats in the first row, $24$ seats in the second row, $28$ seats in the third row, and continues this pattern for a total of $30$ rows.
  • Part A: How many seats are in the 8th row ($t_8$)? [Answer: $48\,\text{seats}$]
  • Part B: How many total seats are in the entire theatre? [Answer: $S_{30} = 2340\,\text{seats}$]

Geometric Series

• A geometric series is defined as the sum of terms in a geometric sequence.
• Example Comparison:
  • Geometric sequence: $3, 6, 12, 24, \dots$
  • Geometric series: $3 + 6 + 12 + 24 + \dots$
  • Illustrative calculation: In the sequence above, the sum of the first four terms ($S_4$) is $45$.

Formula for Determining the Sum of a Geometric Series

In general, the sum $S_n$ of the first $n$ terms in a geometric series is calculated using the following formula:

• Formula: $S_n = \frac{a(r^n - 1)}{r - 1}$
• Constraint: $r \neq 1$
• Variables:
  • $a$: The first term of the series.
  • $n$: The number of terms or the term number.
  • $r$: The common ratio between consecutive terms.
  • $S_n$: The sum of the first $n$ terms in the sequence.

Geometric Series Examples and Applications

Example 1: Analyzing Geometric Series Components

Given the geometric series: $32 - 16 + 8 - \dots + \frac{1}{8}$

• Part A: Determine the total number of terms ($n$) present in the series.
• Part B: Determine the total sum ($S_n$) of this series.

Example 2: Financial Application - Lottery Prize Tiers

A lottery distributes a total of $12$ prizes.

• Prize Distribution:
  • The first ticket drawn receives a prize of $\$10$.
  • Every subsequent ticket receives a prize that is triple the value of the prize immediately preceding it ($r = 3$).
• Part A: Calculate the specific value of the 12th prize ($t_{12}$).
• Part B: Calculate the total amount of prize money distributed across all $12$ winners ($S_{12}$).

Practice Problems for Geometric Series

• Summation Practice: Find $S_{10}$ for the geometric series: $2 - 6 + 18 - 54 \dots$
  • Answer: $88574$
• Biological Application - Fish Hatchery Hatch Rates:
  • At a fish hatchery, eggs fertilized at the same time hatch at different rates. The number of fish that hatched on the first four days was $2$, $10$, $50$, and $250$, respectively.
  • Question: If the current pattern continues, how many fish will have hatched in total after $8$ days?
  • Answer: $195312\,\text{fish}$