Comprehensive Study Guide: Arithmetic and Geometric Series

Course and Institutional Context

Teacher: Eden G. Aniversario, MSc, LPT
Subject: General Mathematics
Institution: Notre Dame of Marbel University, Integrated Basic Education Department, Senior High School
Affiliation: JMJ Marist Brothers, Koronadal City, South Cotabato
Unit Lesson: Lesson 5 - Series

Lesson Objectives

Recitation Objective: Illustrate precisely arithmetic and geometric series through recitation.
Board Work Objective: Represent appropriately series using sigma notation and vice versa through board work.
Practice Exercise Objective: Find correctly the sum of arithmetic and geometric series through practice exercises.
Insight Objective: Value wholeheartedly the lesson through sharing insights.

Real-World Introductory Scenario

Savings Problem: "If you save $\text{₱}50$ today, $\text{₱}100$ tomorrow, then $\text{₱}150$ the next day, how much will you have after $10$ days?"
Mathematical Formulation:
- Day 1 ( $a_1$ ): $\text{₱}50$
- Day 2 ( $a_2$ ): $\text{₱}100$
- Day 3 ( $a_3$ ): $Xa_3 = \text{₱}150\n * Number of days (n $):$ 10\n * Common difference (d $):$ \text{₱}50\n * 10 $\text{th} day savings ($ a_{10} $):$ a_{10} = 50 + (10 - 1)50 = 500\n * Total savings (S_{10} $):$ S_{10} = \frac{10}{2}(50 + 500) = 5(550) = 2750\n * **Total Saved:** \text{₱}2,750\n\n# Fundamentals of Sequences vs. Series\n\n* **Sequence:** An ordered list of numbers.\n* **Series:** The sum of the terms in a sequence.\n* **Finite vs. Infinite Distinction:**\n * **Finite Sequence:** A sequence with a countable number of terms.\n * Example: 2, 4, 6, 8, 10\n * **Infinite Sequence:** A sequence that continues indefinitely without an end.\n * Example: 2, 4, 6, 8, 10, \dots\n * **Finite Series:** The sum of a finite sequence.\n * Example: 2 + 4 + 6 + 8 + 10\n * **Infinite Series:** The sum of an infinite sequence.\n * Example: 2 + 4 + 6 + 8 + 10 + \dots\n\n# Arithmetic Series\n\n* **Definition:** An arithmetic series is the sum of the terms of an arithmetic sequence where consecutive terms differ by a constant common difference d.\n* **Formulas for the Sum S_n of a Finite Arithmetic Series:**\n * **Formula 1 (Using First Term a_1 $and Last Term$ a_n):**\n S_n = \frac{n}{2}(a_1 + a_n)\n * **Formula 2 (Using First Term a_1 $and Common Difference$ d):**\n S_n = \frac{n}{2}[2a_1 + (n - 1)d]\n* **General Term Rule:**\n a_n = a_1 + (n - 1)d\n\n# Geometric Series\n\n* **Definition:** A geometric series is the sum of the terms of a geometric sequence where each term after the first is obtained by multiplying the preceding term by a constant common ratio r\n* **Formula for the Sum S_n of a Finite Geometric Series:**\n S_n = \frac{a_1(1 - r^n)}{1 - r}\n * **Constraint:** r \neq 1\n\n# Worked Examples\n\n* **Example 1 (Arithmetic Series - Formula 1):**\n * **Problem:** Find the sum of the first 10 $terms of the series:$ 2 + 4 + 6 + 8 + \dots\n * **Solution:**\n * a_1 = 2 $,$ d = 2 $,$ n = 10\n * Calculate a_{10} $:$ a_{10} = 2 + (10 - 1)2 = 20\n * Calculate S_{10} $:$ S_{10} = \frac{10}{2}(2 + 20) = 5(22) = 110\n\n* **Example 2 (Arithmetic Series - Formula 2):**\n * **Problem:** Find the sum of the first 10 $terms of the series:$ 2 + 4 + 6 + 8 + \dots\n * **Solution:**\n * S_{10} = \frac{10}{2}[2(2) + (10 - 1)2]\n * S_{10} = 5[4 + (9)2] = 5[4 + 18] = 5(22) = 110\n\n* **Example 3 (Sum of First 1000 Integers):**\n * **Problem:** Find the sum of the first 1,000 positive integers.\n * **Solution:**\n * a_1 = 1 $,$ a_{1000} = 1000 $,$ n = 1000\n * S_{1000} = \frac{1000}{2}(1 + 1000) = 500(1001) = 500,500\n\n* **Example 4 (Geometric Series - Positive Terms):**\n * **Problem:** Find the sum of the first 5 $terms of the given sequence:$ 2, 6, 18, 54, \dots\n * **Solution:**\n * a_1 = 2 $,$ r = \frac{6}{2} = 3 $,$ n = 5\n * S_5 = \frac{2(1 - (3)^5)}{1 - 3} = \frac{2(1 - 243)}{-2} = \frac{2(-242)}{-2} = 242\n\n* **Example 5 (Geometric Series - Alternating Signs):**\n * **Problem:** Find the sum of the first 10 $terms of the geometric sequence:$ -5, 10, -20, 40, -80, \dots\n * **Solution:**\n * a_1 = -5 $,$ r = \frac{10}{-5} = -2 $,$ n = 10\n * S_{10} = \frac{-5(1 - (-2)^{10})}{1 - (-2)} = \frac{-5(1 - 1024)}{1 + 2} = \frac{-5(-1023)}{3} = -5(-341) = 1705\n\n# Practice Exercises\n\n* **Exercise 1 (Arithmetic Series):**\n * **Problem:** Find the sum of the first 12 $terms of$ 5 + 9 + 13 + 17 + \dots\n * **Solution:**\n * Identified sequence parameters: a_1 = 5 $,$ d = 4 $,$ n = 12\n * Find a_{12} $:$ a_{12} = 5 + (12 - 1)4 = 5 + 44 = 49\n * Apply sum formula: S_{12} = \frac{12}{2}(5 + 49) = 6(54) = 324\n\n* **Exercise 2 (Geometric Series):**\n * **Problem:** Find the sum of the first 6 $terms of$ 243, 81, 27, \dots\n * **Identification:** Geometric sequence with a_1 = 243 $,$ r = \frac{81}{243} = \frac{1}{3} $,$ n = 6\n * **Step-by-step Solution:**\n 1. Apply geometric sum formula: S_n = \frac{a_1(1 - r^n)}{1 - r}\n 2. Substitute values: S_6 = \frac{243\left(1 - \left(\frac{1}{3}\right)^6\right)}{1 - \frac{1}{3}}\n 3. Evaluate power: \left(\frac{1}{3}\right)^6 = \frac{1}{729}\n 4. Simplify numerator: 1 - \frac{1}{729} = \frac{728}{729}\n 5. Simplify denominator: 1 - \frac{1}{3} = \frac{2}{3}\n 6. Set up fraction division: S_6 = \frac{243 \times \frac{728}{729}}{\frac{2}{3}}\n 7. Simplify product: Since 243 \times 3 = 729 $,$ 243 \times \frac{728}{729} = \frac{728}{3}\n 8. Divide by denominator: S_6 = \frac{\frac{728}{3}}{\frac{2}{3}} = \frac{728}{3} \times \frac{3}{2} = \frac{728}{2} = 364\n\n# Summation Notation (Sigma Notation)\n\n* **Definition:** Sigma notation uses the upper-case Greek letter sigma \sum (meaning "sum") to express a series in a compact form.\n* **Anatomy of Sigma Notation:**\n \sum_{n=1}^{5} 2n\n * **n:** Index of summation.\n * **1:** Lower limit of summation (starting term index).\n * **5:** Upper limit of summation (ending term index).\n * **2n $:** Explicit formula for the$ n\text{th} term.\n * **Reading:** "The sum from 1 to 5 of 2n".\n\n* **Deriving General Rule for Sigma Notation:**\n * Sequence: 2 + 4 + 6 + 8 + 10\n * First term a_1 = 2 $, common difference$ d = 2\n * Formula: a_n = a_1 + (n - 1)d = 2 + (n - 1)2 = 2 + 2n - 2 = 2n\n * Expression: \sum_{n=1}^{5} 2n\n\n* **Summation Notation for Infinite Series:**\n * Infinite Series: 2 + 4 + 6 + 8 + 10 + \dots\n * Representation: \sum_{n=1}^{\infty} 2n\n * Explanation: Since the series is infinite, the upper limit is replaced by infinity (\infty).\n\n* **Converting Series to Sigma Notation:**\n * **Problem:** Write 25 + 50 + 75 + \dots + 250 using summation notation.\n * **Solution:**\n * a_1 = 25 $,$ d = 25\n * a_n = 25n\n * Find number of terms n $:$ 25n = 250 \implies n = 10\n * Answer: \sum_{n=1}^{10} 25n\n\n* **Evaluating Sigma Notation Expression:**\n * **Problem:** Find the sum of the series \sum_{k=5}^{10} (k^2 + 1)\n * **Solution:**\n * For k = 5 $:$ 5^2 + 1 = 26\n * For k = 6 $:$ 6^2 + 1 = 37\n * For k = 7 $:$ 7^2 + 1 = 50\n * For k = 8 $:$ 8^2 + 1 = 65\n * For k = 9 $:$ 9^2 + 1 = 82\n * For k = 10 $:$ 10^2 + 1 = 101\n * Total Sum: 26 + 37 + 50 + 65 + 82 + 101 = 361\n\n# Exit Questions & Core Insights\n\n* **Question 1:** What is the difference between arithmetic and geometric series?\n * *Answer:* An arithmetic series has a constant common difference (d $) added between successive terms, whereas a geometric series has a constant common ratio ($ r) multiplied between successive terms.\n* **Question 2:** How to find the sum of arithmetic and geometric series?\n * *Answer:* \n * For arithmetic series: Use S_n = \frac{n}{2}(a_1 + a_n) $or$ S_n = \frac{n}{2}[2a_1 + (n - 1)d].\n * For geometric series: Use S_n = \frac{a_1(1 - r^n)}{1 - r} $, provided$ r \neq 1\n\n# Values Formation Integration\n\n* Learning arithmetic and geometric series builds **perseverance**, **discipline**, and **critical thinking**.\n* Solving step-by-step mathematical problems teaches patience and planning.\n* Recognizing patterns demonstrates how consistent actions—such as daily saving—lead to meaningful financial results and long-term financial responsibility.\n\n# Academic Reference\n\n* Asuncion, V.J. & Ramo, M.N. (2021). *Business mathematics* (1st ed.). Quezon City: Tech Factors Inc.\n\n# Comprehensive Assessment Quiz\n\n* **Item 1:** What is the sum of the first 5 $terms of the arithmetic series:$ 4 + 8 + 12 + \dots?\n * a. 40\n * b. 60\n * c. 70\n * d. 80\n * *Solution:* a_1 = 4 $,$ d = 4 $,$ n = 5 $.$ S_5 = \frac{5}{2}[2(4) + (5-1)4] = \frac{5}{2}[8 + 16] = \frac{5}{2}(24) = 60\n * *Answer:* b. 60\n\n* **Item 2:** What is the 6 $\text{th} term of the geometric series:$ 3 + 6 + 12 + 24 + \dots?\n * a. 96\n * b. 48\n * c. 192\n * d. 120\n * *Solution:* a_1 = 3 $,$ r = 2 $,$ n = 6 $.$ a_6 = a_1 r^{n-1} = 3(2^5) = 3(32) = 96\n * *Answer:* a. 96\n\n* **Item 3:** What is the sum of the first 10 $terms of the arithmetic series:$ 5 + 8 + 11 + \dots?\n * a. 135\n * b. 185\n * c. 165\n * d. 150\n * *Solution:* a_1 = 5 $,$ d = 3 $,$ n = 10 $.$ S_{10} = \frac{10}{2}[2(5) + (10-1)3] = 5[10 + 27] = 5(37) = 185\n * *Answer:* b. 185\n\n* **Item 4:** Find the sum of the arithmetic series: 10 + 13 + 16 + \dots + 40\n * a. 204\n * b. 196\n * c. 180\n * d. 210\n * e. None of the above\n * *Solution:* a_1 = 10 $,$ d = 3 $,$ a_n = 40 $.$ 40 = 10 + (n-1)3 \implies 30 = 3(n-1) \implies n = 11 $.$ S_{11} = \frac{11}{2}(10 + 40) = \frac{11}{2}(50) = 275\n * *Answer:* e. None of the above\n\n* **Item 5:** What is the sum of the first 6 $terms of a geometric sequence where$ a = 2 $and$ r = 3?\n * a. 728\n * b. 1456\n * c. 782\n * d. 820\n * e. None of the above\n * *Solution:* S_6 = \frac{2(1 - 3^6)}{1 - 3} = \frac{2(1 - 729)}{-2} = 728\n * *Answer:* a. 728\n\n* **Item 6:** What is the value of the sum: \sum_{n=1}^{4} (2n + 1)?\n * a. 20\n * b. 24\n * c. 18\n * d. 22\n * e. None of the above\n * *Solution:* n=1: 3 $,$ n=2: 5 $,$ n=3: 7 $,$ n=4: 9 $. Sum$ = 3 + 5 + 7 + 9 = 24\n * *Answer:* b. 24\n\n* **Item 7:** An arithmetic series has first term 7 $and common difference$ 5 $. What is the$ 10\text{th} term?\n * a. 47\n * b. 52\n * c. 57\n * d. 62\n * *Solution:* a_{10} = 7 + (10 - 1)5 = 7 + 45 = 52\n * *Answer:* b. 52\n\n* **Item 8:** Which of the following is not a geometric sequence?\n * a. 2, 4, 8, 16\n * b. 5, 10, 20, 40\n * c. 3, 6, 12, 25\n * d. 1, 3, 9, 27\n * *Solution:* For option c, \frac{6}{3} = 2 $and$ \frac{12}{6} = 2 $, but$ \frac{25}{12} \neq 2. It does not have a constant common ratio.\n * *Answer:* c. 3, 6, 12, 25\n\n* **Item 9:** What is the sum of the first 100 natural numbers?\n * a. 5,000\n * b. 4,950\n * c. 5,050\n * d. 4,990\n * *Solution:* S_{100} = \frac{100}{2}(1 + 100) = 50(101) = 5050\n * *Answer:* c. 5,050\n\n* **Item 10:** What is the value of the series \sum_{k=1}^{4} k?\n * a. 10\n * b. 15\n * c. 6\n * d. 20\n * *Solution:* 1 + 2 + 3 + 4 = 10\n * *Answer:* a. 10\n\n* **Item 11:** What does the expression \sum_{n=1}^{5} (2n + 1) represent?\n * a. 5 $terms starting from$ 1 $to$ 5\n * b. Sum of the first 5 odd numbers\n * c. An arithmetic series with common difference 1\n * d. A geometric sequence\n * *Solution:* Evaluating for n=1 $to$ 5 $yields the sequence of values$ 3, 5, 7, 9, 11, which corresponds to 5 terms calculated index-by-index from 1 to 5.\n * *Answer:* a. 5 $terms starting from$ 1 $to$ 5\n\n* **Item 12:** Evaluate \sum_{n=0}^{3} 3^n\n * a. 39\n * b. 40\n * c. 41\n * d. 27\n * *Solution:* 3^0 + 3^1 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40\n * *Answer:* b. 40\n\n* **Item 13:** How many terms are in the series \sum_{n=3}^{7} n?\n * a. 4\n * b. 5\n * c. 6\n * d. 7\n * *Solution:* Number of terms = \text{Upper Limit} - \text{Lower Limit} + 1 = 7 - 3 + 1 = 5\n * *Answer:* b. 5\n\n* **Item 14:** What type of sequence is represented by \sum_{n=1}^{\infty} 3n?\n * a. Arithmetic\n * b. Geometric\n * c. Harmonic\n * d. Quadratic\n * *Solution:* The sequence generated by 3n $is$ 3, 6, 9, 12, \dots $, which has a common difference of$ 3, making it arithmetic.\n * *Answer:* a. Arithmetic\n\n* **Item 15:** What is the 7 $\text{th} term of the arithmetic series where the first term is$ 6 $and the common difference is$ 3?\n * a. 24\n * b. 21\n * c. 18\n * d. 15\n * *Solution:* a_7 = 6 + (7 - 1)3 = 6 + 18 = 24\n * *Answer:* a. 24$$