Arithmetic sequence, Exercises of Mathematics

Download Arithmetic sequence and more Exercises Mathematics in PDF only on Docsity! Lesson 4 - Arithmetic Sequence Lesson Objectives: * Determines n" term of an arithmetic sequence ¢ Solve problems in finding the sum of the terms of a given arithmetic sequence Activity 1: Engage A child building a tower with blocks uses 15 for the bottom row. Each row has 2 fewer blocks than the previous row. Suppose that there are 8 rows in the tower. From the given situation, how can we determine the number of blocks used in the 6" row, 7" row or 8" row without writing the arithmetic sequence one by one? How can we get the total number of blocks used in building the tower? Given; a=15 n= 6th, 7th, 8th d=2 an = 6th, 7th, 8th To determine the number of blocks used in the 6th row, 7th row, or 8th row without writing the arithmetic sequence one by one | need to use the formula of the arithmetic sequence a, = a,+ (n- 1) d. Where an is the nth term, aris the first term, n is the (term position) or also known as the number of terms and d is the common difference. In getting the total number of blocks used in building the tower. | need to use the formula of the arithmetic series which is n Sn, 2 [2a1, (n — 1)d] . Using those specified formula will help us solve the problem on knowing the total numbers of blocks in the different rows of the tower. As well as the total number of blocks used in building the tower. COE-05-12;Module /6" January 2020 Determining the number of blocks used in the 6th row. Determining the number of blocks used in the 7th row. COE-05-12;Module /6" January 2020 Getting the total number of blocks used in building the tower. Activity 2:Explore! In order to solve the problem in Activity 1, you need to identify first the essential given values that will be used. These values are to be substituted into the formula in finding any term or also known as the n" term in an arithmetic sequence and also in finding the sum of an arithmetic sequence. In answering the following process questions, refer to the given problem in Activity 1. COE-05-12;Module /6" January 2020 Process Questions: a) What is the first term in the sequence? The first term in the sequence is 15. b) What is the term you want to find? The term | want to find are the 6‘ , 7‘ and 8" row. c) What is the term position from the sequence? The term position from the sequence is 8", What is the common difference between terms? The common difference between terms is -2. d) Is identifying the given values in a problem important in solving the problem? Why? Yes, because the values given in every problem is essential in order unlock or to define the values that we are going to find. In that way, we will be able to solve the problem. So, when you don’t identify the given values, the problem wouldn't be solve. Activity 3: Clear Up Finding the nth Term of an Arithmetic Sequence If you wish to find any term (also known as the n" term) in the arithmetic sequence, the arithmetic sequence formula should help you to do so to make it easier and faster. The critical step is to be able to identify or extract Known values from the problem that will eventually be substituted into the formula itself. Let's start by examining the essential parts of the formula: Parts of the Arithmetic Sequence Formula term position | a, =a, +(n-l)d n” term t common difference | first term Me nye mee re ee Ter Ny cr emt mene Se (Source:https:/www.chilimath.com/lessons/intermediate-algebra/arithmetic-sequence-formula/) where: COE-05-12;Module /6" January 2020 n= the n® term in the sequence ai = the first term in the sequence n= the term position (ex: for 6" term, n = 6) d =the common difference between terms Let's put this formula in action! Example 1: Find the 35" term in the arithmetic sequence 3, 9, 15, 21, ... There are three things needed in order to find the 35" term using the formula: «the first term (@1) « the common difference between consecutive terms (qd) * and the term position (n) From the given sequence, we can easily read off the first term and common difference. The term position is just the n value in the n® term, thus in the 35" term, n=35. first term 3, 9, 15, 21... 6 6 6 Therefore, the known values that will be used in substituting in the arithmetic formula are firstterm= A; =3 common difference = d =6 term position = N =35 So the solution to finding the missing term is, An = A1 + (n - 1)d Ags = 3 + (35-1)-6 =3+(34)-6 =3+204 Ags = 207 Thus, the 35" term in the arithmetic sequence is 207. COE-05-12;Module /6" January 2020 Arithmetic Series An arithmetic series is the sum of an arithmetic sequence. Formulas for Arithmetic Series: i applicable onlyif the rl” term S,= 3 (4 +a,) <——— orlast term is present in the sequence S,=F[24 +(n-1)4] where @, = the first term An = the n® term n= the number of terms d =the common difference Let's put this formula in action! Example 1: Find the sum of the first 20 terms of the arithmetic sequence 3, 7, 11, ... This arithmetic sequence has the first term @1 = 3, and a common difference of 4. To determine the value of n, refer to the number of terms given in the problem. Thus, the n value would be n = 20. The sum of the arithmetic sequence is calculated as, Sn “3 Poy (n — 1)d] 5, = (203) + (20-1)-4] = 10 [6 + (19)-4] =10(6 + 76) = 10 (82) Sn = 820 Therefore, the sum of the first 20 terms of the arithmetic sequence is 820. Example 2: Find the sum of the arithmetic sequence 10, 20, 30, ..., 1000. This arithmetic sequence has the first term @1 = 10, An = 1000 and a common n 2 (a1 + an) difference of 10. Thus, we can use the first formula Sp * . The missing COE-05-12;Module /6" January 2020 term that we need to identify is the value of n. To find n, use the explicit formula for an arithmetic sequence which is An = A1 + (n—1)d. In finding n, we have: An = @1+(n-1)d 1000 = 10 + (n—1)-10 1000 = 10 + 10n— 10 10n = 1000 10n 1000 10-10 n=100 The value of n is 100. Therefore, the essential values are now complete that will be substituted in the formula. So the solution in getting the sum of the arithmetic sequence is, n S72 (a; + an) in 100 g, 2g (10 + 1000) Sp = 50 (1010) Sp = 50,500 Therefore, the sum of the arithmetic sequence is 50,500. Practice Exercise: a) Consider the arithmetic sequence 2, 5, 8, 11, ... Find the sum of the first 15 terms. ___345 COE-05-12;Module /6" January 2020 b) Determine the sum of the arithmetic series 3, 8, 13, ..., 73. ___ 570 COE-05-12;Module /6" January 2020 1) How many blocks are used in 7" row? 3 COE-05-12;Module /6" January 2020 2) How many blocks are used in building the tower? 64 n *3 [2a, Formula ; Sn + (n - 1)d] There are 64 blocks used in building the tower. COE-05-12;Module /6" January 2020 Activity 5: Assessment Direction: Make your own word problem in real life situation that applies the concept in finding n* term of an arithmetic sequence or finding the sum of the terms of a given arithmetic sequence. Your score will be based in the given rubric. e My own word problem in real life situation that applies the concept in finding n* term of an arithmetic sequence. | want to buy a cellphone that worth of 6,000 pesos. but the average amount that | can save from my part time job salary is only 200 pesos. Within the first week, | have saved 1.400 pesos. How much would | save in the next 9th week? Will | be able to purchase the phone? d=200 €@1= 1400 ag = n=9th Formula to be use ; An = Ait (n-1)d Answer 1 Solution; An = @1+(n-1)d Ag = 1400 + (9 — 1) 200 pg = 1400 + (8) 200 pg = 1400 + 1600 A= 3000 | will not be able to purchase the phone since the money that | will be save in the next 9th week is only 3,000 pesos. COE-05-12;Module /6" January 2020