ACTIVITY SHEET IN MATHEMATICS 10 NAME: ______________________________________ SCORE:______________ GRADE LEVEL:___________________ WEEK: 1 Task 1: Tell whether the given numbers form a sequence or not. 1. 6, 13 15 17 2 2 2 , 7, , 8, ,….. 2. 12, 13, 17, 19, 22, …. 3. 63, 127, 255, 511, 1023,….. 4. 28, 31, 34, 37, 40, …. 1 3 1 4 7 1 2 3 5 2 2 2 5. , , 3, 5, , …… 6. − , 1, − , 2, − , 3, …….. 7. -3, -1, 0, 4, 5, ….. 1 1 2 1 8. -2, − , 0, , , ,…….. 2 4 3 8 9. 1, , , , 4 5 2 5 , …… 5 5 17 13 5 7 9 10. 2, , 3, , 4, , …. 2 2 2 Task 2: Describe the pattern formed. Find the next three terms. 1. 1, 4, 7, 10, …….. 2. 4, -8, 16, -32, ……. 1 1 1 3. , , , …….. 2 4 8 4. 0, 3, 7, 12, 18 …… 5. 1, 3, 9 ………. Task 3: In a Physics experiment, a ball is dropped at the height of 24 feet. For each time it bounces, the ball reaches half its previous height. Below is the table showing the height of the ball bounces until it comes to rest. What will be the height of the ball if it bounces on the 4th ,5th, 6th and 7th time? No. of 1 2 3 4 5 6 7 Bounces Height 12 6 3 (ft) _____________________ Math Teacher Parent’s Name/Signature ACTIVITY SHEET IN MATHEMATICS 10 NAME: ______________________________________ SCORE:______________ GRADE LEVEL:___________________ WEEK: 2 Task 1: Tell whether the following sequence form an arithmetic sequence . 1. 12, 17, 22, 27, 32, …… 2. 15, 11, 7, 3, -1, ….. 3. 1, 1, 2, 3, 5, 8 4. 4, 16, 64, …… 5. 100, 50. 25, 25/2… 6. -3, -5, -7, -9, …. 7. 8, -4, 2, -1, … 8. 5, 5.5, 6, 6.5 …… 9. 20, 17, 14, 10, …. 10. 11, 16, 21, 27, ……. Task 2A: Find the next 5 terms of each of the given arithmetic sequences. 1. 2. 3. 4. 5. 3,6,9,12…….. 5,9, 13,……. 21, 15, 9 …. 2, -3, -8, …. 11, 8, 5…… Task 2B: Solve the following arithmetic sequences and name the first 4 terms. 1. 𝑎1 = 4, 𝑑 = 3 2. 𝑎1 = 7, 𝑑 = 5 3. 𝑎1 = −8, 𝑑 = 3 1 1 2 2 4. 𝑎1 = , 𝑑 = 5. 𝑎1 = 6, 𝑑 = −2 Task 3: Solve each problem. (Show your solution on the space provided) 1. What is the 12th term of the arithmetic sequence 5, 8, 11, 14,…? 2. What is the common difference of an arithmetic sequence if the first term is 12 and the 10th term is 39? 3. The given sequence -37, -33, -29, …form an arithmetic sequence, what will be the 10th term of the sequence? 4. Michael deposited Php20,000 on an investment that will give Php1,750 for every year that his money stays in the account. How much money will he have in his account by the end of year 10? 5. Sarah works in a law firm with a beginning salary of Php450,000 per year. If her salary will be increased yearly by Php2,000, what would be her salary on the 10th year of working? _____________________ Math Teacher Parent’s Name/Signature ACTIVITY SHEET IN MATHEMATICS 10 NAME: ______________________________________ SCORE:______________ GRADE LEVEL:___________________ WEEK: 3-4 Drill: 1. What is the 10th term of the sequence 2, 5, 8….? 2. What is the common difference of the sequence in #1? 3. How can we tell if the given the sequence is an arithmetic sequence? Explain your answer. _____________________________________ The teacher will give a preliminary problem and the students will be asked to see the difference from the previous lesson discussed to them. This will lead the students to the next topic. GEAR UP: Find the next 4 terms. 1. 3, 6, 12, 24, ………. 2. 1, 3, 9, 27 ……….. 3. ½, ¼,1/8, ……. 4. 80,40, 20…….. 5. 400,200,100…. Questions: 1. Hoe did you arrive with your answer? 2. What operations were used? 3. What makes the sequence different from the previous lesson discussed? KEY IDEA A geometric sequence or 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. If the common ratio is greater than 1, the sequence grows at an increasing rate. Thus, the common ratio is known as the growth factor. If the common ratio is less than 1, the sequence shrinks at a decreasing rate. Thus, this common ratio is known as the decay factor. Let Us Practice! In the given example: 2, 6, 18, 54, 162, . . . , the common ratio is 3. If you present the first term as t1, using the pattern above, you have, 1st term = t1 2nd term = t2 3rd term = t3 th 4 term = t4 5th term = t5 6th term = t6 Therefore, organizing the pattern, you have the number of terms and expressions for the nth term as follows: 1st term = t1 2nd term = t2 3rd term = t3 th 4 term = t4 5th term = t5 6th term = t6 nth term = tn 2 · 3 0 = t1 2 · 31 = t1 r1 2 · 32 = t1r2 3 2 · 3 = t1r3 2 · 34 = t1 r4 2 · 35 = t1 r5 2 · 3n-1 = t1 rn-1 You will notice that the exponent of the common ratio is 1 less than the number of terms you are looking for. So, if you are looking the 3rd term, the exponent of the common ratio is 2; for the 4th term, the exponent of the common ratio is 3; for the 6th term, the common ratio is 5, etc… Therefore, for the nth term, the exponent of the common ratio is n – 1, where n is the number of terms. To find the nth term of a geometric sequence, the formula is tn = t1 rn-1 where n = the number of terms r = common ratio t1 = first term tn = last term So, if you want to find the 10th term of the sequence 2, 6, 18, 54, 162, . . . , then you have to use the formula for convenience tn = t1 rn-1. First identify the given before you substitute to the formula: t1 = 2 r =3 n = 10 tn = ? first term common ratio number of terms the term you are looking for Solution: tn = t1 rn-1 t10 = 2 (3)10 -1 t10 = 2 (3)9 t10 = 2 (19,683) t10 = 39,366 Substitute the given value Simplify the exponent Simplify(3)9 . It is equal to 19, 683 Therefore the 10th term of the sequence 2, 6, 18, 54, 162, . . . is 39,366. LET US PRACTICE! A. Find the next four terms of the sequence. 1. 1 1 , , 1, 2, . . . 4 2 2. - 1 1 , , -1, 2, . . 4 2 3. 32, -16, 8, -4, . . . 4. - 1 1 1 1 ,- ,- ,- , .. 32 16 8 4 5. 2 , 2, 2 2 , 4, . . . 6. 1 1 1 , , , 1, . . . 27 9 3 7. -1, 4, -16, 32, . . . 8. 12, 24, 48, 96, . . . 9. 10, 2, 2 2 , ,... 5 25 10. 7, 14, 28, 56, . . . LET US DO IT!! Tell whether or not each number sequence is a geometric sequence. Those that are geometric, give the common ratio. a. 6, 24, 96, 384, . . . b. 5, -10, 20, -40, . . . c. 1 , 1, 2, 4, . . . 2 d. 3, 9, 27, . . . e. 2, 4, 6, 8, 10, . . . f. 1 ,1, 3, 9, . . . 3 g. 7, 9, 11, 13, . . . h. 15, 5, 5 5 , ,... 3 9 1 ,... 2 i. 4, 2, 1, j. 3, 7, 11, 15, . . . Given the following sequences, tell which are geometric and which are arithmetic sequences. State their common ratio or common difference. k. 5, 20, 80, 320, . . . l. 1, 4, 9, 16, . . . m. 3, 8, 13, 18, . . . n. 1, 4, 9, 16, . . . o. 31, 38, 45, 52, . . . p. 4, 12, 36, 108, . . . q. 1, -3, 5, -7, . . . r. 1, 6, 36, 216, . . . s. 1 2 3 4 , , , ,... 2 3 4 5 2 3 4 5 , , , ,... 1 4 9 16 B. Write the first five terms of the geometric progression where t. 1. a1 = 2, 2. a1 = 3, r=3 r=2 1 3. a1 = 10, r = 2 1 4. a1 = 32, r = 4 5. a1 = 3, r = -2 6. a1 = 2, r = -3 2 1 7. a1 = , r = 3 2 8. a1 = 1, r = 0.5 9. a1 = -1, r = 0.5 10. a1 = -2, r = -2 _____________________ Math Teacher Parent’s Name/Signature Name: _________________________________ Grade and Section: ______________ School: __________________________________ Date: _____________________ LEARNING ACTIVITY SHEET Mathematics 10 Week 6 – 1st Quarter This learning activity sheet provides varied activities that will help you solve problems involving arithmetic and geometric sequences. At the end of this learning activity sheet, the learners are expected to: solve problems involving sequences. (M10AL–If–2) Directions: Using the nth term of the arithmetic and geometric sequence, solve the following items. Write your answer on the space provided. 1. The 20th term of the arithmetic sequence 5, 13, 21,29, 37,… 2. The 15th term of an arithmetic sequence whose a1 = 12 and whose d = - 9. 3. The 13th term of the geometric sequence whose first term is 3 and whose common ratio is -2. 4. The 8th term of the geometric sequence whose a1 = -4 and whose r = 3. 5. The 24th term of the arithmetic sequence whose first term is 25 and whose common difference is -6. Directions: Decode the tagline set by the Department of Health in line with its program “Fight CoViD – 19” by solving the following problems involving sequences. Write the word that corresponds to each answer on the space provided below to complete the tagline. at Flatten Always 6 15 days 13 days Stay to curve 2 7 78732 the circle home 60 6132 3 A country struck by the Corona Virus Disease – 19 (CoViD19) reached its peak infecting 81,920 people in total. It was found that the virus spread is in a Geometric Progression and that the number of people infected gets doubled each day reason for its government to implement a total lockdown. If there were only 5 people infected on the first day, how many days had passed to reach its peak? 1 2 After a knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes for the first week. Each week thereafter, he suggests you to increase the time by 6 minutes. How many minutes will you need after the 9th week? 3 A microbiologist was able to isolate 12 endophytic bacterial strains from nipa palm in Bulacan, Philippines. In this study, screening of endophytic bacteria examined their multiple plant growth-promoting traits. These bacteria are able to increase their numbers by the process of replication called binary fission whereby their population triples every generation time. Find the total population (sum) of bacteria after 9 generation time? 4 The common ratio on item number 1. 5 The common difference on item number 2. 6 The common ratio of item number 3. Tagline: 1 20 30 40 50 60 Directions: Solve the following problems involving sequences. Write your answer on the space provided. *Show your complete solution. 1. An auditorium has 10 seats in the first row and 4 additional seats in each successive row. How many seats will be there on the 18th row? 2. Kristine used her savings to buy a new car amounting to Php720,000.00. Suppose a car’s value depreciates yearly at 10%, what is her car’s value after 4 years? 3. A side of an apartment building is shaped like a steep staircase. The windows are arranged in columns. The first column has 2 windows, the second column has 5 windows, the next has 8, then 11, and so on. How many windows are on the side of the apartment if there are 15 columns? 4. Assuming that the geometric sequence continues, what is the height of a bouncing ball on its 8th bounce if its initial height after the first drop is 140 ft., next is 70 ft., and so on? 5. Determine the seating capacity of a gymnasium with 18 seats on the first row, 20 on the second, 22 on the third. The sequence of the seating capacity per row continues until the 14th row. Solution: Reference/s: Callanta, Melvin; Canonigo ,Allan, et. al.: Grade 10 Mathematics Learner’s Module. Edited by Maxima Acelajado, Ph. D. Pasig City, Philippines. DEPED-IMCS Rex Book Store, Inc. 2015. _____________________ Math Teacher Parent’s Name/Signature
