Sequence After going through this lesson, you are expected to generate patterns and sequences. Probably without even knowing it, you have been observing and creating patterns ever since you were a very small child. You probably made repeating patterns with shapes, such the one below with triangles, circles, and squares. Definition. A sequence is a set of numbers written in a specific order: a1, a2, a3, a4, a5, a6,…, an. The number a1 is called the 1st term, a2 is the 2nd term, and in general, an is the nth term. A sequence having a finite number of terms is called a finite sequence. A sequence having a infinite number of terms is called a infinite sequence. Worksheet 1: Worksheet 1: B. Find the next three terms of the given sequences. 1. 2, 5, 8, 11, __, __, __ 6. 2, 4, 8, 16, __, __, __ 2. 1, -3, -7, -11, __, __, __ 7. 2, 6, 18, 54, __, __, __ 3. 1, 4, 16, 64, __, __, __ 8. 3.2, 4.3, 5.4, __, __, __ 4. 60, 48, 36, 24, __, __, __ 9. -7,-9, -11, -13, __, __, __ 5. 1, 4, 9, 16, 25, __, __, __ 10. 7, 4, 1,-2 , __, __, __ Worksheet 1: C. Write the first three terms of the sequence whose nth term is given by the rule. 1. an =2n-1 2. an =12-3n2 3. an = -5n3 + 9n2 - 4n + 1 Arithmetic Sequence Definition: An arithmetic sequence is a sequence where each succeeding term is obtained by adding a fixed number. The fixed number is called the common difference which is denoted as d. To find the next terms in an arithmetic sequence, we use the formula: ππ = π1 + (π − 1)π where; ππ – the last nth term π1 – the first term π – the number of terms in the sequence Study the given examples below and then identify if it is arithmetic or not. 1. 10, 13, 16, 19,… 2. 2, 6, 18, 54… 3. 57, 49, 41 Definition: The terms between any two nonconsecutive terms of an arithmetic sequence are known as arithmetic means. Answer Learning Task 2A on page 10 Assignment: Use intermediate papers as your answer sheets. • Answer Learning Task 2B and 2C on page 10. • Answer Learning Task 3 on page 10. Worksheet 2 A. Determine whether the given sequence is arithmetic or not. Write ‘A’ for arithmetic and ‘NA’ for not. 1. 25, 21, 17, 13, 9,... 2. 5, 7, 9, 11, 13, ... 3. 6, 10, 14, 18, 22, ... 4. 1, 3, 9, 11, 13, ... 5. -1, 4, 9, 14, 19, ... 6. -14, -7, 0, 7, 14, ... 7. 6 7 8 9 , , , , 2, . . . 5 5 5 5 8. 6.2, 6.7, 7.2, 7.7, 8.2, ... 9. 1, 1 1 1 1 , , , , 2 3 4 5 ... 10. 5, 9, 13, 17, 21, ... B. Answer each problem. 1. Insert three arithmetic means between 11 and 35. 2. Find the 30th term of the sequence, 15, 19, 23, 27, 31, ... . 3. Find the sum of the first 15 terms of the sequence, 15, 19, 23, 27, 31,... Geometric Sequence A Geometric Sequence, also known as Geometric Progression, is a set of terms in which each term after the first is obtained by multiplying the preceding term by the same fixed number called the Common Ratio which is commonly represented by r. Examples: 1. 6, 18, 54, 162, ... 2. 4, -8, 16, -32, ... 3. 5, 30, 180, 1080, ... 4. 2, 1, 5. 1 1 , , 2 4 ... 1 1 , , 1, 2, . . . 4 2 Answer Learning Task 1 on page 12 Geometric Sequence The formula for the nth term of a geometric sequence is ππ = π1 β π π−1 Examples: 1. Write the first four terms of the geometric sequence whose nth term is given by ππ = 3(−5)π−1 . 2. Find the 8th term of the geometric sequence 24, 12, 6, 3, ... 3. Find the 11th term of the geometric sequence, 6, 18, 54, 162. 4. Find the 9th term of the geometric sequence 4, -8, 16, 32, ... How to Use the Scientific Calculator? Geometric Means The terms between two given terms in a geometric sequence is called geometric means. Examples: • Insert two geometric means between 6 and 750. • Given −4, π2 , π3 , 500 as consecutive terms of a geometric sequence, find π3 . 1 2 • The first term of a geometric sequence is and the 4th term is 4 . 27 What are the second and third terms? Solve each problem. 1. Insert 3 geometric means between 120 and 15 . 2 2. Given −6, π2 , π3 , 384 as consecutive terms of a geometric sequence, find π3 . Finite Geometric series It is an expression formed by adding the terms of a geometric sequence. Example: The geometric series of the geometric sequence 11, 22, 44, 88 is 165. 11 + 22 + 44 + 88 = 165 General Formula: The πth partial sum of a finite geometric series with π ≠ π is given by ππ π − ππ πΊπ = π−π Examples: • The geometric series of the geometric sequence 11, 22, 44, 88 is 165. • Find the sum of the first 12 terms of the geometric sequence 3, -9, 27, -81, 243, ... . • Find the sum of the first 10 terms of a geometric 1 sequence with the first term is 12, and common ratio, . 2 Infinite Geometric Series It is an infinite series whose successive terms have a common ratio. The series converges if and only if the absolute number of the common ratio is less than one |π| < 1 . Formula: π1 π∞ = 1−π Examples: • Find the sum of the infinite geometric sequence, 1 1 1 1 1, , , , , . . . 2 4 8 16 • Find the sum of the infinite geometric series with π1 = 3 168 and π = . 4 Worksheet 3. A. State whether the given sequence is arithmetic, geometric or neither. If it is arithmetic sequence, give the common difference. If it is geometric, give the common ratio. 1. 11, 14, 17, 20, ... 2. 4, 8, 16, 32, ... 3. 5, 8, 12, 17, 26, ... 4. 32, 28, 24, 20, ... 5. 1, 8, 27, 64, ... 6. 7, 10, 15, 22, 31, ... 7. 1, -3, 5, -7, ... 8. 80, 40, 20, 10, ... 9. 20, 30, 36, 42, ... 10. 100, -50, 25, -12.5, ... Worksheet 3 B. Find the indicated term of each geometric sequence. 1. a1 = 7, r = 3, a10 = ? 2. a1 = 6, r = 1/2, a9 = ? 3. a1 = 10, r = 0.02, a11 = ? C. Answer the following problems. Show your solution. 1. Insert 2 geometric means between 5 and 1080. 2. Find the sum of the first 10 terms of the geometric sequence 100, 20, 4, 0.8, ... . Worksheet 3 Answer the following. 1. Find the sum of the first 10 terms of the geometric sequence 8 16 32 64 4, , , , , . . . . 3 9 27 81 2. Find the sum of the infinite geometric series with π1 = 1 32 and π = − . 8 Application of Sequences Review: Arithmetic and Geometric Sequence Answer Learning Task 1 on page 15 Word Problems Suppose the auditorium of the Tagaytay International Convention Center (TICC) has 20 seats in the first row and that each row has 2 more seats than the previous row. If there are 30 rows in the auditorium, how many seats are in the last row? 1. Joan started a new job with an annual salary of Php 150,000 in 2007. If she receives a Php 12,000 raise each year, how much will her annual salary be in 2017? 2.Micheal saved Php 50 in January. Suppose he saved twice the amount he saved the previous month. How much would he save at the end of one year? More problems... Worksheet 5. Answer each problem. 1. Ric bought a machine worth Php 29,000. If it depreciates in value by Php 500 each year, how much is it worth at the end of 10 years? 2. Find the seating capacity of a movie house with 40 rows of seats if there are 25 seats in the first row, 28 seats in the second row, 31 in the third row, and so on. 3. A car that cost Php 700,000 depreciates 15% in value each year for the first five years. What is its cost after 5 years? A. Determine if the sequence is arithmetic or geometric. 1. -2, -4, -8, -16, ... 2. 65, 60, 55, 50, 45, ... 3. 8, 13, 18, 23, 28, ... 4. 1, 1.5, 2.25, 3.375, ... B. Determine if the sequence is arithmetic or geometric, and then find the given term. 5. 11th term: 5, 3, 1, –1, … ____________ 6. 23rd term: 0.1, 0.15, 0.2, 0.25, … _____________ 7. 6th term: 25, 75, 225, 675, … ______________ C. Answer each problem. (5pts. each) 1. Find a2 in the geometric sequence, 4, a2, 484. 2. How many numbers between 7 and 501 are divisible by 5? 3. A gardener makes a triangular planting with 40 plants in the front row, 36 in the second row, 32 in the third row, and so on. If the pattern is consistent, how many plants will there be in the 8th row? Quiz 1 - G10 A. Determine if the sequence is arithmetic or geometric. 1. 100, -50, 25, -12.5 ... 2. 4, 13 14 , , 3 3 5,... 3. an= −4 (−3)n − 1 4. an=16 + 3n B. Determine if the sequence is arithmetic or geometric, and then find the given term. 5. 22nd term: –2, –5, –8, –11, –14, … ____________ 6. 10th term: a1 = 320, r = 0.5 ____________ 7. 50th term: –9, 2, 13, 24, 35, … ______________ C. Answer each problem. (5pts. each) 1. Suppose a ball is dropped from a height of 16 feet and it bounces up to 75% of its previous height after each bounce. Find the height of the eight h bounce. 2. Find the sum of the first 100 positive odd integers. 3. Insert two arithmetic means between 21 and 33.
